{"id":15124,"date":"2019-12-27T16:37:07","date_gmt":"2019-12-27T11:07:07","guid":{"rendered":"https:\/\/kseebsolutions.guru\/?p=15124"},"modified":"2021-07-02T15:34:56","modified_gmt":"2021-07-02T10:04:56","slug":"1st-puc-maths-question-bank-chapter-11","status":"publish","type":"post","link":"https:\/\/kseebsolutions.guru\/1st-puc-maths-question-bank-chapter-11\/","title":{"rendered":"1st PUC Maths Question Bank Chapter 11 Conic Sections"},"content":{"rendered":"<p>Students can Download Maths Chapter 11 Conic Sections Questions and Answers, Notes Pdf, <a href=\"https:\/\/kseebsolutions.guru\/1st-puc-maths-question-bank\/\">1st PUC Maths Question Bank with Answers<\/a> helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.<\/p>\n<h2>Karnataka 1st PUC Maths Question Bank Chapter 11 Conic Sections<\/h2>\n<p>Question 1.<br \/>\nName the different conics.<br \/>\nAnswer:<br \/>\nCircle, parabola, ellipse and hyperbola.<\/p>\n<p><strong>Circles<\/strong><\/p>\n<p>Question 2.<br \/>\nDefine a circle and derive the equation of the circle whose centre and radius is given.<br \/>\nAnswer:<br \/>\nDefinition: A circle is a set of all points in a plane that are equidistant from a fixed point in a plane. OR A circle is the locus of a point which moves in a plane such that its distance from a fixed point in the plane is constant.<br \/>\nNote:<\/p>\n<ul>\n<li>Fixed point is called centre.<\/li>\n<li>The distance from the centre to a point on the circle is called radius.<\/li>\n<\/ul>\n<p>Equation of a circle, given its centre and radius:<br \/>\nLet C(h,k) be the centre y and r, the radius of the circle. Let P(x,y) be any point on the circle. Then, by definition, |CPl = r<br \/>\n\u2234 By distance formula, we have<br \/>\n\\(\\sqrt{(x-h)^{2}+(y-k)^{2}}=r\\)<br \/>\n\\(\\text { i.e., }(x-h)^{2}+(y-k)^{2}=r^{2}\\)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74292\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-1.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 1\" width=\"170\" height=\"196\" \/><br \/>\nNote: If the centre is at the origin ,then h = 0,k = 0. Therefore the equation of the circle with centre at the origin and radius r is x<sup>2<\/sup> + y<sup>2<\/sup> = r<sup>2<\/sup><\/p>\n<p>Question 3.<br \/>\nFind the equation of the circle with<br \/>\n(i) Centre (0,0) and radius r<br \/>\n(ii) Centre (0,2) and radius 2<br \/>\n(iii) Centre (-2,3) and radius 4<br \/>\n(iv) Centre (-3,2) and radius 4.<br \/>\n(v) Centre (1,1) and radius \\(\\sqrt{2}\\)<br \/>\n(vi) \\(\\text { Centre }\\left(\\frac{1}{2}, \\frac{1}{4}\\right)\\) and radius \\(\\frac{1}{12}\\)<br \/>\n(viii) Centre (-a,-b) and radius \\(\\sqrt{a^{2}-b^{2}}\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74293\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-2.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 2\" width=\"396\" height=\"413\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-2.png 396w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-2-288x300.png 288w\" sizes=\"(max-width: 396px) 100vw, 396px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74294\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-3.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 3\" width=\"372\" height=\"596\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-3.png 372w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-3-187x300.png 187w\" sizes=\"(max-width: 372px) 100vw, 372px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 4.<br \/>\nWrite the standard form (General form) of a circle.<br \/>\nAnswer:<br \/>\nStandard form of a circle is x<sup>2<\/sup> + y<sup>2<\/sup> + 2 gx + 2 fy + c = 0<br \/>\n(\u2235 in the above problems, we observe that the equations of the circles are of the form<br \/>\nx<sup>2<\/sup> + y<sup>2<\/sup> + ( )x + ( )y + () = 0<\/p>\n<p>Note:<\/p>\n<ul>\n<li>Centre = (-g, -f)<br \/>\n\\(\\text { Radius }=\\sqrt{g^{2}+f^{2}-c}\\)<\/li>\n<li>The co-efficients of x<sup>2<\/sup> and y<sup>2<\/sup> are equal to unity.<\/li>\n<li>There is no term containing the product xy.<\/li>\n<li>If g<sup>2<\/sup> + f<sup>2<\/sup> -c &gt; 0, then the circle is real circle.<\/li>\n<li>If g<sup>2<\/sup> + f<sup>2<\/sup> &#8211; c = 0, then the circle is a point circle.<\/li>\n<li>If g<sup>2<\/sup> + f<sup>2<\/sup> &#8211; c&lt; 0, then the circle is imaginary circle.<\/li>\n<li>If the centre of the circle is on the x-axis, then equation of the circle is<br \/>\nx<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + c = 0 and conversely. (Here f = 0)<\/li>\n<li>If the centre of the circle is on the y-axis, then the equation of circle is<br \/>\nx<sup>2<\/sup> + y<sup>2<\/sup> + 2fy + c = 0 and conversely. (Here g = 0)<\/li>\n<li>In the equation, x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0, if c = 0, then the circle passes through the origin and circle equation is x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy = 0<\/li>\n<\/ul>\n<p>Question 5.<br \/>\nFind the centre and radius of the circles.<br \/>\n(i) x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c = 0<br \/>\n(ii) x<sup>2<\/sup> + y<sup>2<\/sup> + 8x + 10y -8 = 0<br \/>\n(iii) x<sup>2<\/sup> + y<sup>2<\/sup> &#8211; 4x &#8211; 8y &#8211; 45 = 0<br \/>\n(iv) x<sup>2<\/sup> + y<sup>2<\/sup> &#8211; 8x &#8211; 10y -12 = 0<br \/>\n(v) (x + 5)<sup>2<\/sup> + (y &#8211; 3)<sup>2<\/sup> = 36<br \/>\n(vi) 2x<sup>2<\/sup> + 2y<sup>2<\/sup>-x = 0<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74295\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-4.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 4\" width=\"390\" height=\"621\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-4.png 390w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-4-188x300.png 188w\" sizes=\"(max-width: 390px) 100vw, 390px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74296\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-5.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 5\" width=\"372\" height=\"616\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-5.png 372w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-5-181x300.png 181w\" sizes=\"(max-width: 372px) 100vw, 372px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74297\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-6.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 6\" width=\"372\" height=\"493\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-6.png 372w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-6-226x300.png 226w\" sizes=\"(max-width: 372px) 100vw, 372px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74298\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-7.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 7\" width=\"377\" height=\"476\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-7.png 377w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-7-238x300.png 238w\" sizes=\"(max-width: 377px) 100vw, 377px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 6.<br \/>\nFind the equation of the circle passing through the points (2, -2) and (3, 4) and whose centre lies on the line x + y = 2<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74299\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-8.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 8\" width=\"392\" height=\"550\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-8.png 392w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-8-214x300.png 214w\" sizes=\"(max-width: 392px) 100vw, 392px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74300\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-9.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 9\" width=\"376\" height=\"116\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-9.png 376w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-9-300x93.png 300w\" sizes=\"(max-width: 376px) 100vw, 376px\" \/><\/p>\n<p>Question 7.<br \/>\nFind the equation of the circle<br \/>\n(i) passing through the points (4, 1) and (6, 5) and whose centre lies on 4x + y = 16.<br \/>\n(ii) Passing through the points (2, 3) and (-1, 1) and centre is on the line x &#8211; 3y -11 = 0.<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74301\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-10.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 10\" width=\"394\" height=\"373\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-10.png 394w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-10-300x284.png 300w\" sizes=\"(max-width: 394px) 100vw, 394px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74302\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-11.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 11\" width=\"398\" height=\"489\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-11.png 398w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-11-244x300.png 244w\" sizes=\"(max-width: 398px) 100vw, 398px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74303\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-12.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 12\" width=\"348\" height=\"599\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-12.png 348w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-12-174x300.png 174w\" sizes=\"(max-width: 348px) 100vw, 348px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 8.<br \/>\nFind the equation f the circle with radius 5 whose centre lies on x-axis and passing through the point (2,3).<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74304\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-13.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 13\" width=\"391\" height=\"435\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-13.png 391w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-13-270x300.png 270w\" sizes=\"(max-width: 391px) 100vw, 391px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74305\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-14.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 14\" width=\"296\" height=\"125\" \/><\/p>\n<p>Question 9.<br \/>\nFind the equation of the circle passing through (0, 0) and making intercepts a and b on the coordinate axes.<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74306\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-15.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 15\" width=\"409\" height=\"494\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-15.png 409w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-15-248x300.png 248w\" sizes=\"(max-width: 409px) 100vw, 409px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74307\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-16.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 16\" width=\"254\" height=\"360\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-16.png 254w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-16-212x300.png 212w\" sizes=\"(max-width: 254px) 100vw, 254px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74308\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-17.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 17\" width=\"404\" height=\"561\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-17.png 404w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-17-216x300.png 216w\" sizes=\"(max-width: 404px) 100vw, 404px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 10.<br \/>\nFind the equation of the circle with centre (2,2) and passes through the point (4,5).<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74309\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-18.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 18\" width=\"348\" height=\"472\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-18.png 348w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-18-221x300.png 221w\" sizes=\"(max-width: 348px) 100vw, 348px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74310\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-19.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 19\" width=\"347\" height=\"289\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-19.png 347w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-19-300x250.png 300w\" sizes=\"(max-width: 347px) 100vw, 347px\" \/><\/p>\n<p>Question 11.<br \/>\nDoes the point (-2.5, 3.5) lie inside, outside or on the circle x<sup>2<\/sup> + y<sup>2<\/sup> = 25?<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74311\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-20.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 20\" width=\"343\" height=\"354\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-20.png 343w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-20-291x300.png 291w\" sizes=\"(max-width: 343px) 100vw, 343px\" \/><\/p>\n<p><strong>Parabola:<\/strong><\/p>\n<p>Question 1.<br \/>\nDefine parabola.<br \/>\nAnswer:<br \/>\nA parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane. OR Parabola is the locus of a point which moves in a plane such that its distance from a fixed point is equal to its distance from a fixed line. The fixed point is called focus and fixed line is called directrix of the parabola.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74312\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-21.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 21\" width=\"305\" height=\"164\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-21.png 305w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-21-300x161.png 300w\" sizes=\"(max-width: 305px) 100vw, 305px\" \/><\/p>\n<p>Question 2.<br \/>\nWrite the standard equations of parabola and their respective graphs<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74313\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-22.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 22\" width=\"411\" height=\"670\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-22.png 411w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-22-184x300.png 184w\" sizes=\"(max-width: 411px) 100vw, 411px\" \/><br \/>\nNote: Parabolas are symmetrical about co-ordinate axes.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 3.<br \/>\nDerive the equation of a parabola in the form y<sup>2<\/sup> =4ax<br \/>\nAnswer:<br \/>\nLet F be the focus and l be the directrix. Draw FM perpendicular to l. Take the\u00a0 mid-point of FM as origin O and introduce the co\u00adordinate axes shown in the figure. Let the distance between the directrix and focus be 2a. (i.e., FM = 2a) Then |OF| = |OM| = a and F = (a,0) and equation of directrix is x = -a i.e., x + a =0.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74314\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-23.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 23\" width=\"179\" height=\"166\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74315\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-24.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 24\" width=\"388\" height=\"383\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-24.png 388w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-24-300x296.png 300w\" sizes=\"(max-width: 388px) 100vw, 388px\" \/><\/p>\n<p>Question 4.<br \/>\nDefine latus rectum of a parabola and find the length of the latus rectum of the parabola y<sup>2<\/sup> = 4<br \/>\nAnswer:<br \/>\nDefinition: Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola.<br \/>\nLength of latus rectum of y<sup>2<\/sup> = 4ax:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74316\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-25.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 25\" width=\"203\" height=\"203\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-25.png 203w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-25-150x150.png 150w\" sizes=\"(max-width: 203px) 100vw, 203px\" \/><br \/>\nBy definition of parabola, AF = AC . But<br \/>\nFM = 2a = AC<br \/>\n\u2234\u00a0 AF &#8211; 2a<br \/>\nSince the parabola is symmetric with respect to x-axis then<br \/>\nAF = FB = 2a<br \/>\nBy definition, AB = latus rectum<br \/>\n\u2234 AB = AF + FB = 2a + 2a = 4a<br \/>\n\u2234\u00a0 Length of latus rectum = 4a.<\/p>\n<p>Note: Observations: (Useful for solving exercise) Vertex = (0, 0)<\/p>\n<table width=\"674\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\" width=\"78\">Equation<\/td>\n<td style=\"text-align: center;\" width=\"50\">Axis<\/td>\n<td style=\"text-align: center;\" width=\"58\">Focus<\/td>\n<td style=\"text-align: center;\" width=\"71\">Directrix<\/td>\n<td style=\"text-align: center;\" width=\"38\">LR<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"78\">y<sup>2<\/sup> = 4ax<\/td>\n<td style=\"text-align: center;\" width=\"50\">x-axis<\/td>\n<td style=\"text-align: center;\" width=\"58\">(a,0)<\/td>\n<td style=\"text-align: center;\" width=\"71\">x + a = 0<\/td>\n<td style=\"text-align: center;\" width=\"38\">4a<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"78\">y<sup>2<\/sup> =-4ax<\/td>\n<td style=\"text-align: center;\" width=\"50\">x-axis<\/td>\n<td style=\"text-align: center;\" width=\"58\">(-a, 0)<\/td>\n<td style=\"text-align: center;\" width=\"71\">x-a = 0<\/td>\n<td style=\"text-align: center;\" width=\"38\">4a<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"78\">x<sup>2<\/sup> = 4ay<\/td>\n<td style=\"text-align: center;\" width=\"50\">y-axis<\/td>\n<td style=\"text-align: center;\" width=\"58\">(0 ,a)<\/td>\n<td style=\"text-align: center;\" width=\"71\">y + a = 0<\/td>\n<td style=\"text-align: center;\" width=\"38\">4 a<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"78\">x<sup>2<\/sup> &#8211; 4ay<\/td>\n<td style=\"text-align: center;\" width=\"50\">y-axis<\/td>\n<td style=\"text-align: center;\" width=\"58\">(0 -a)<\/td>\n<td style=\"text-align: center;\" width=\"71\">y &#8211; a = 0<\/td>\n<td style=\"text-align: center;\" width=\"38\">4 a<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 5.<br \/>\nFind the coordinates of the focus axis of the parabola, equation of the directrix and length of latus rectum.<br \/>\n(i) y<sup>2<\/sup> = 8x<br \/>\n(ii) y<sup>2<\/sup> =12x<br \/>\n(iii) y<sup>2<\/sup> = 10x<br \/>\n(iv) y<sup>2 <\/sup>= &#8211; 8x<br \/>\n(v) x<sup>2 <\/sup>\u00a06y<br \/>\n(vi) x<sup>2 <\/sup>= &#8211; 16y<br \/>\n(vii)\u00a0 x<sup>2<\/sup> = -9y<br \/>\n(viii) 3y<sup>1<\/sup> = 12x<br \/>\n(ix)\u00a0\u00a0 3x<sup>2<\/sup> = -4y<br \/>\n(x) 2y<sup>2<\/sup> = -6x (viii, ix, x are not in exercise)<br \/>\nAnswer:<br \/>\nGiven (i) y<sup>2<\/sup> = 8x. This is of the form<br \/>\ny<sup>2<\/sup> = 4 ax<br \/>\n\u2234 4a = 8 \u21d2 = 21<br \/>\nWe have, Focus: (a,0) = (2,0)<br \/>\nAxis: x-axis<br \/>\nEquation of directive: x + a = 0 i.e., x + 2=0<br \/>\nLength of latus rectum: 4a = 4(2) =8<\/p>\n<p>(ii) Given: y<sup>2<\/sup> = 12x<br \/>\nForm: y<sup>2 <\/sup>= 4ax<br \/>\n\u2234 4a = 12 \u21d2 a = 3<br \/>\nWe have,<br \/>\nFocus : (a,0)=(3,0)<br \/>\nAxis : x &#8211; axis<br \/>\nDirectrix: x + a=0 i.e., x + 3 = O<br \/>\nLatus rectum: 4a = 4(3) = 12<\/p>\n<p>(iii) Try yourself.<br \/>\n\\(F=\\left(\\frac{5}{2}, 0\\right), x \\text { -axis, } x+\\frac{5}{2}=0,10\\)<\/p>\n<p>(iv) Given y<sup>2 <\/sup> = &#8211; 8x<br \/>\nForm: y<sup>2 <\/sup> = &#8211; 4ax<br \/>\n4a = 8 \u21d2 a=2<br \/>\nWe have,<br \/>\nFocus: ( &#8211; a,0) = ( &#8211; 2,0)<br \/>\nAxis: x-axis<br \/>\nDirectrix : x &#8211; a = 0 \u21d2 x + 2 = 0<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74317\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-26.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 26\" width=\"255\" height=\"539\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-26.png 255w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-26-142x300.png 142w\" sizes=\"(max-width: 255px) 100vw, 255px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74318\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-27.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 27\" width=\"320\" height=\"568\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-27.png 320w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-27-169x300.png 169w\" sizes=\"(max-width: 320px) 100vw, 320px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74319\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-28.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 28\" width=\"292\" height=\"398\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-28.png 292w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-28-220x300.png 220w\" sizes=\"(max-width: 292px) 100vw, 292px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 6.<br \/>\nFind the equation of the parabola that satisfies the give conditions.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74320\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-29.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 29\" width=\"306\" height=\"223\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-29.png 306w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-29-300x219.png 300w\" sizes=\"(max-width: 306px) 100vw, 306px\" \/><\/p>\n<p>(I)<br \/>\n(i) Vertex (0, 0) passing through (2, 3) and axis is along x-axis.<br \/>\n(ii) Vertex (0, 0) passing through (5, 2) and symmetrix with respect toy &#8211; axis.<br \/>\n(iii) Symmetric about y-axis and passing through the point (2, -3).<br \/>\nAnswer:<br \/>\nI. (i) Given that<br \/>\nFocus = (2, 0), Directrix : x = &#8211; 2<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74321\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-30.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 30\" width=\"229\" height=\"169\" \/><br \/>\nSince (2, 0) lies on the positive then axis of the required parabola is x-axis!<br \/>\n\u2234 By data, required parabola is<br \/>\ny<sup>2 <\/sup>&#8211; 4ax &#8230;&#8230;&#8230;&#8230; (1)<br \/>\nWe have, focus = (a,0) = (2,0) (given)<br \/>\n\u2234 a = 2<br \/>\n\u2234 (1) = y<sup>2 <\/sup>= 4(2)x<br \/>\n\u21d2 y<sup>2 <\/sup> = 8x\u00a0 is the required parabola.<\/p>\n<p>(ii) y2 = 24x (Try yourself)<\/p>\n<p>(iii) Given, F= (0,- 3), l : y =3<br \/>\nSince F lies on the negative y-axis then axis of the parabola is y-axis.<br \/>\nBy data, required parabola is<br \/>\nx<sup>2 <\/sup>= &#8211; 4ay &#8230;&#8230;&#8230;&#8230;. (1)<br \/>\nBut F = (0, &#8211; 3) = (0, &#8211; a)<br \/>\n\u2234 a = 3<br \/>\n(1) = x<sup>2 <\/sup>= -4(3)y \u21d2 x<sup>2 <\/sup>= &#8211; 12y<br \/>\nis the required equation of the parabola.<\/p>\n<p>(II)<br \/>\n(i) Given that, vertex = (0, 0) and focus = (0, 2)<br \/>\nSince focus lies on the positive y-axis then the required equation of the parabola is<br \/>\nx<sup>2 <\/sup>= &#8211; 4ay &#8230;&#8230;&#8230;&#8230;&#8230;&#8230;.(1)<br \/>\nSince, focus = (0,2) = (0,a) \u2234 a = 2<br \/>\n\u2234 (1) \u21d2 x<sup>2 <\/sup>= &#8211; 4(2)y<br \/>\n\u21d2 x<sup>2 <\/sup>= &#8211; 8y is required equation<\/p>\n<p>(ii) Given that, vertex = (0, 0) and focus = (3,0)<br \/>\nSince focus lies on the positive x-axis, then the required equation of the parabola is<br \/>\ny<sup>2<\/sup> =4ax &#8230;&#8230;&#8230;&#8230;&#8230;&#8230;. (1)<br \/>\nSince focus = (3, 0) = (a, 0) a = 3<br \/>\n(1) \u21d2 y<sup>2<\/sup> = 4(3)x \u21d2 y<sup>2<\/sup> = 12x<\/p>\n<p>(iii) Given that, vertex = (0, 0) and focus = (-2, 0)<br \/>\nSince focus lies on the negative x-axis, then the required equation of the parabola is<br \/>\ny<sup>2<\/sup> = -4ax &#8230;&#8230;&#8230;&#8230;&#8230;&#8230;. (1)<\/p>\n<p>Since focus = (-2, 0) = (-a, 0) a- 2<br \/>\n(1) \u21d2 y<sup>2<\/sup> = -4(2)x =5.1<br \/>\ny<sup>2<\/sup> = -8x<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>(III)<br \/>\n(i) Given that, Axis: x &#8211; axis<br \/>\nVertex = (0, 0) and point = (2, 3)<br \/>\nSince (2, 3) lies in the first quadrant and axis is x-axis, then the required equation of the parabola is<br \/>\ny<sup>2<\/sup> =4ax &#8230;&#8230;&#8230;&#8230;&#8230;&#8230;. (1)<br \/>\nSince (1) passes through the point (2, 3) then, we have<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74322\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-31.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 31\" width=\"215\" height=\"164\" \/><\/p>\n<p>(ii) Given: V = (0, 0), Axis: y-axis, point = (5, 2) Since the point (5, 2) lies in the first quadrant and axis is y-axis then, the required equation of the parabola is<br \/>\nx<sup>2<\/sup> =4ay &#8230;&#8230;&#8230;&#8230;&#8230;&#8230;. (1)<br \/>\nSince (1) passing through the point (5, 2) then, we have<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74323\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-32.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 32\" width=\"240\" height=\"168\" \/><\/p>\n<p>(iii) Given: Axis: y-axis, point = (2,-3)<br \/>\nSince (2, -3) lies in the fourth quadrant and axis is y-axis then, the required equation of the parabola is x<sup>2 <\/sup>= -4ay &#8230;&#8230;&#8230;&#8230;&#8230;&#8230;. (1)<br \/>\nSince (1) passing through the point (2, -3) then, we have<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74324\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-33.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 33\" width=\"273\" height=\"235\" \/><\/p>\n<p><strong>Ellipse<\/strong><\/p>\n<p>Question 1.<br \/>\nDefine ellipse.<br \/>\nAnswer:<br \/>\nAn ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant. The two fixed points are called foci.<\/p>\n<p>Points to be noted:<\/p>\n<ul>\n<li>The mid point of the line segment joining the foci is called centre of the ellipse.<\/li>\n<li>The line segment through the foci is called the major axis.<\/li>\n<li>The line segment through the centre and perpendicular to the major axis is called minor axis.<\/li>\n<li>The ends points of major axis are called vertices of the ellipse.<\/li>\n<li>Length of major axis is 2a<\/li>\n<li>length of minor axis is 2b<\/li>\n<li>Distance between the foci is 2c<\/li>\n<li>Relationship between semi-major axis, semi\u00adminor axis and distance of the focus from the centre of the ellipse is a<sup>2<\/sup> =b<sup>2<\/sup> + c<sup>2<\/sup> i.e., \\(c=\\sqrt{a^{2}-b^{2}}\\)<\/li>\n<\/ul>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 2.<br \/>\nDefine eccentricity of an ellipse.<br \/>\nAnswer:<br \/>\nThe eccentricity of an ellipse is the ratio of the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse. Eccentricity is denoted by \\(e.e=\\frac{c}{a} \\)(Here c &lt;a \u2234 e&lt; 1)<\/p>\n<p>Question 3.<br \/>\nWrite standard equations of an ellipse and their respective graphs.<br \/>\nAnswer:<br \/>\nStandard equations of an ellipse are\u00a0 \\(\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74325\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-34.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 34\" width=\"400\" height=\"201\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-34.png 400w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-34-300x151.png 300w\" sizes=\"(max-width: 400px) 100vw, 400px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74326\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-35.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 35\" width=\"403\" height=\"161\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-35.png 403w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-35-300x120.png 300w\" sizes=\"(max-width: 403px) 100vw, 403px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 4.<br \/>\nDerive the equation of an ellipse in the form<br \/>\n\\(\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\)<br \/>\nAnswer:<br \/>\nLet F<sub>1<\/sub> and F<sub>2<\/sub> be the<br \/>\nfoci and \u2018O\u2019 be the midpoint of F<sub>1<\/sub>F<sub>2<\/sub><br \/>\nLet \u2018O\u2019 be the origin and line from \u2018O\u2019 &lt;\u00adthrough F<sub>1<\/sub> and F<sub>2<\/sub> be x-axis and through \u2018O\u2019 perpendicular to the x-axis be the y-axis.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74327\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-36.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 36\" width=\"188\" height=\"201\" \/><br \/>\nLet F<sub>1<\/sub> =(c,0) and F<sub>2<\/sub> =(-c,0)<br \/>\nLet P(x, y) be any point on the ellipse such that the sum of the distances from P to the two foci be \u20182a\u2019. i.e., PF<sub>1<\/sub> + PF<sub>1<\/sub>=2a &#8230;&#8230;&#8230;&#8230;&#8230;&#8230;&#8230; (1)<br \/>\nBy distance formula, we have<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74328\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-37.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 37\" width=\"415\" height=\"541\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-37.png 415w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-37-230x300.png 230w\" sizes=\"(max-width: 415px) 100vw, 415px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74329\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-38.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 38\" width=\"123\" height=\"147\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 5.<br \/>\nDefine latus rectums of an ellipse and find the length of the latus rectum of the ellipse<br \/>\n\\(\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\)<br \/>\nAnswer:<br \/>\nDefinition: Latus rectum of an ellipse is a line segment perpendicular to Q the major axis through any of the foci and whose ends points lie on the ellipse. In the figure, AB and CD are latus rectum. Let the length of AF<sub>1<\/sub> be l . Then A = (c,l)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74330\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-39.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 39\" width=\"173\" height=\"194\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74331\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-40.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 40\" width=\"354\" height=\"657\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-40.png 354w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-40-162x300.png 162w\" sizes=\"(max-width: 354px) 100vw, 354px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74332\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-41.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 41\" width=\"286\" height=\"264\" \/><\/p>\n<p>Question 6.<br \/>\nFind the coordinates of foci, the vertices, length of major and minor axes, the eccentricity and the latus rectum of the ellipse.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74333\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-42.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 42\" width=\"351\" height=\"306\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-42.png 351w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-42-300x262.png 300w\" sizes=\"(max-width: 351px) 100vw, 351px\" \/><br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74334\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-43.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 43\" width=\"301\" height=\"471\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-43.png 301w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-43-192x300.png 192w\" sizes=\"(max-width: 301px) 100vw, 301px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74335\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-44.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 44\" width=\"322\" height=\"407\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-44.png 322w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-44-237x300.png 237w\" sizes=\"(max-width: 322px) 100vw, 322px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74336\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-45.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 45\" width=\"308\" height=\"708\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-45.png 308w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-45-131x300.png 131w\" sizes=\"(max-width: 308px) 100vw, 308px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74337\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-46.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 46\" width=\"289\" height=\"534\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-46.png 289w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-46-162x300.png 162w\" sizes=\"(max-width: 289px) 100vw, 289px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74338\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-47.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 47\" width=\"297\" height=\"556\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-47.png 297w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-47-160x300.png 160w\" sizes=\"(max-width: 297px) 100vw, 297px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74339\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-48.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 48\" width=\"312\" height=\"354\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-48.png 312w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-48-264x300.png 264w\" sizes=\"(max-width: 312px) 100vw, 312px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74340\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-49.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 49\" width=\"387\" height=\"750\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-49.png 387w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-49-155x300.png 155w\" sizes=\"(max-width: 387px) 100vw, 387px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74341\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-50.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 50\" width=\"301\" height=\"647\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-50.png 301w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-50-140x300.png 140w\" sizes=\"(max-width: 301px) 100vw, 301px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 7.<br \/>\nFind the equation for the ellipse that satisfies the given conditions:<br \/>\n(i) Vertices (\u00b15,0), foci (\u00b14,0)<br \/>\n(ii) Vertices (0,\u00b113), foci (0,\u00b15)<br \/>\n(iii) Vertices (\u00b113,0), foci (\u00b15,0)<br \/>\n(iv) Ends of major axis (\u00b13,0) ends of minor axis (0,\u00b12)<br \/>\n(v) Ends of major axis \\((0, \\pm \\sqrt{5})\\), ends of minor axis (\u00b11,0)<br \/>\n(vi) Length of major axis 26, foci (\u00b15,0)<br \/>\n(vii) Length of major axis is 20, foci (0,\u00b15)<br \/>\n(viii) Length of minor axis is 16, foci (0,\u00b16)<br \/>\n(ix) Foci (\u00b13,0), a = 4<br \/>\n(x) b\u00a0= 3, c = 4 centre at the origin; foci on the x-axis.<br \/>\n(xi) Centre (0, 0), major axis on the y-axis and passes through the point (3,2) and (1,6)<br \/>\n(x) Major axis on the x-axis and passes through the points (4,3) and (6,2).<br \/>\nAnswer:<br \/>\n(i) Given vertices Vertices (\u00b15, 0), foci (\u00b14,0)<br \/>\nWe have, vertices = (\u00b1a,0) = (\u00b15,0) a = 5<br \/>\nAnd foci = (\u00b1c, 0) (\u00b14,0) c = 4<br \/>\nWe have a<sup>2<\/sup> =b<sup>2<\/sup> + c<sup>2<br \/>\n<\/sup>i.e., b<sup>2<\/sup> =a<sup>2<\/sup> -c<sup>2<\/sup><br \/>\n=(5)<sup>2<\/sup> -(4)<sup>2<\/sup> =25-16 = 9<br \/>\ni.e., b<sup>2<\/sup> = 9 \u21d2 b = \u00b13<br \/>\n\u2234 a = 5 and b = 3<br \/>\n\u2234 Required equation of ellipse is<br \/>\n\\(\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 \\quad( \\text { foci lie on } x \\text { -axis })\\)<br \/>\n\\(\\text { i.e., } \\sqrt{\\frac{x^{2}}{25}+\\frac{y^{2}}{9}=1}\\)<\/p>\n<p>(ii) Given Vertices = (0,\u00b113), foci = (0,\u00b15)<br \/>\nSince foci lie on y-axis, then required equation of the ellipse is \\(\\frac{x^{2}}{b^{2}}+\\frac{y^{2}}{a^{2}}=1\\) &#8230;&#8230;&#8230;&#8230;(1)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74342\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-51.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 51\" width=\"301\" height=\"299\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-51.png 301w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-51-150x150.png 150w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-51-300x298.png 300w\" sizes=\"(max-width: 301px) 100vw, 301px\" \/><\/p>\n<p>(iii) Given : Vertices = (\u00b113,0), foci= (\u00b15,0)<br \/>\nSince foci lie on the x-axis, then required equation of the ellipse is<br \/>\n\\(\\frac{x^{2}}{b^{2}}+\\frac{y^{2}}{a^{2}}=1\\)<br \/>\nSince, vertices = (\u00b1a, 0) = (\u00b113,0) \u2234 a =13<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74343\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-52.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 52\" width=\"294\" height=\"315\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-52.png 294w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-52-280x300.png 280w\" sizes=\"(max-width: 294px) 100vw, 294px\" \/><\/p>\n<p>(iv) Given: Ends of major axis (\u00b13,0) ends of minor axis (0,\u00b12)<br \/>\ni.e., (\u00b1 a,0) = (\u00b13, 0) and (0, \u00b1b) = (0,\u00b12)<br \/>\n\u2234 a = 3 and b = 2<br \/>\nRequired equation of the ellipse is<br \/>\n\\(\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 \\text { i.e., } \\frac{x^{2}}{9}+\\frac{y^{2}}{4}=1\\)<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>(v) Given: Ends of major axis \\((0, \\pm \\sqrt{5})\\), ends of minor axis (\u00b11,0)<br \/>\nSince ends of major axis lie on the y-axis, then required equation of the ellipse is,<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74344\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-53.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 53\" width=\"336\" height=\"196\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-53.png 336w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-53-300x175.png 300w\" sizes=\"(max-width: 336px) 100vw, 336px\" \/><\/p>\n<p>(vi) Given: Length of major axis 26, foci (\u00b15,0) Since, foci lie on the x-axis, t hen required equation of the<br \/>\n\\(\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\)<br \/>\nWe have, length of major axis =2a = 26 (given)<br \/>\nand foci = (\u00b1c, 0) = (\u00b15,0)<br \/>\n\u2234a=\\(\\frac{26}{2}=13\\)<br \/>\nand foci = (\u00b1c, 0) = (\u00b15,0)<br \/>\n\u2234 c = 5<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74345\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-54.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 54\" width=\"319\" height=\"178\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-54.png 319w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-54-300x167.png 300w\" sizes=\"(max-width: 319px) 100vw, 319px\" \/><\/p>\n<p>(vi) Given: Length of minor axis is 16, foci (0,\u00b16)<br \/>\nSince, foci lie on the y-axis, then required equation of the ellipse is<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74346\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-55.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 55\" width=\"294\" height=\"398\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-55.png 294w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-55-222x300.png 222w\" sizes=\"(max-width: 294px) 100vw, 294px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>(viii) Given: Length of minor axis is 16, foci (0,\u00b16)<br \/>\nSince, foci lie on the y-axis, then required equation of the ellipse is<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74347\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-56.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 56\" width=\"335\" height=\"337\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-56.png 335w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-56-150x150.png 150w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-56-298x300.png 298w\" sizes=\"(max-width: 335px) 100vw, 335px\" \/><\/p>\n<p>(ix) Given: Foci (\u00b13,0), a = 4<br \/>\nSince foci lie on the x-axis, then the required equation of the ellipse is<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74348\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-57.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 57\" width=\"367\" height=\"261\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-57.png 367w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-57-300x213.png 300w\" sizes=\"(max-width: 367px) 100vw, 367px\" \/><\/p>\n<p>(x) Given: b = 3, c = 4 a = (0,0)<br \/>\nSince foci on the x-axis, then required equation of the ellipse is,<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74349\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-58.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 58\" width=\"310\" height=\"180\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-58.png 310w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-58-300x174.png 300w\" sizes=\"(max-width: 310px) 100vw, 310px\" \/><\/p>\n<p>(xi) Given: Centre (0, 0), axis : y-axis<br \/>\nSince major axis is on the y-axis then, required equation of the ellipse is,<br \/>\n\\(\\frac{x^{2}}{b^{2}}+\\frac{y^{2}}{a^{2}}=1\\)<br \/>\nSince (1), passes through the point (3, 2) and (1,6) then, we<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74350\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-59.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 59\" width=\"362\" height=\"454\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-59.png 362w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-59-239x300.png 239w\" sizes=\"(max-width: 362px) 100vw, 362px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74351\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-60.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 60\" width=\"348\" height=\"647\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-60.png 348w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-60-161x300.png 161w\" sizes=\"(max-width: 348px) 100vw, 348px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p><strong>Hyperbola:<\/strong><\/p>\n<p>Question 1.<br \/>\nDefine hyperbola.<br \/>\nAnswer:<br \/>\nA hyperbola is the set of all points in a plane, the difference of whose distance from two fixed points in the plane is a constant.<\/p>\n<p>Note: Observations<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74352\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-61.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 61\" width=\"345\" height=\"195\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-61.png 345w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-61-300x170.png 300w\" sizes=\"(max-width: 345px) 100vw, 345px\" \/><br \/>\n(i) The two fixed points F<sub>1<\/sub>, F<sub>2<\/sub> are called foci of the hyperbola.<br \/>\n(ii) The mid-point of the line segment joining the foci is called centre of the hyperbola.<br \/>\n(iii) The line through foci is called transverse axis and the line through the centre and perpendicular to the transverse axis is called conjugate axis<br \/>\n(iv) The points at which the hyperbola intersects the transverse axis are called vertices of the hyperbola.<br \/>\n(v) Distance between foci is denoted by 2c, distance between vertices by 2a, and define the quantity b as \\(b=\\sqrt{c^{2}-a^{2}} \\text { i.e., } \\sqrt{b^{2}=c^{2}-a^{2}}\\)<br \/>\n(vi) Length of transverse axis = 2a<br \/>\nLength of conjugate axis = 2b<br \/>\n(vii) Eccentricity = \\(e=\\frac{c}{a}(\\text { Here } c \\geq a)\\)<br \/>\n\u2234 Eccentricity is never less than 1.<\/p>\n<p>Standard equation of hyperbola and their graphs:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74353\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-62.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 62\" width=\"373\" height=\"685\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-62.png 373w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-62-163x300.png 163w\" sizes=\"(max-width: 373px) 100vw, 373px\" \/><\/p>\n<p>Question 2.<br \/>\nDerive the standard equation of the hyperbola in the form<br \/>\n\\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74354\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-63.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 63\" width=\"285\" height=\"154\" \/><\/p>\n<p>Let F<sub>1<\/sub> and F<sub>2<\/sub> be the foci and \u2018O\u2019 be the midpoint of line segment F<sub>1<\/sub>F<sub>2<\/sub>.<br \/>\nLet \u2018O\u2019 be the origin and the line through \u2018O\u2019, F<sub>1<\/sub> and F<sub>2<\/sub> be the x-axis. The line through O, perpendicular to the x-axis be the y-axis.<br \/>\nLet F<sub>1<\/sub>= (c,0) and F<sub>2<\/sub> = (-c,0)<br \/>\nLet P = (x,y) be any point on the hyperbola such that the difference of the distances from P to the<br \/>\nF<sub>1<\/sub> and F<sub>2<\/sub> be 2a.<br \/>\ni.e., PF<sub>2<\/sub>-PF<sub>1<\/sub>=2a<br \/>\nUsing the distance formula, we have<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74355\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-64.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 64\" width=\"391\" height=\"570\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-64.png 391w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-64-206x300.png 206w\" sizes=\"(max-width: 391px) 100vw, 391px\" \/><br \/>\nLength of latus rectum \\(=\\frac{2 b^{2}}{a}\\)<br \/>\nNote: A hyperbola in which a = b is called an equilateral hyperbola.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 3.<br \/>\nDefine latus rectum of hyperbola.<br \/>\nAnswer:<br \/>\nLatus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola.<\/p>\n<p>Question 4.<br \/>\nFind the coordinates of the foci, vertices, eccentricity, length of transverse axis and conjugate axis and length of latus rectum of the hyperbola.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74357\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-66.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 66\" width=\"336\" height=\"192\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-66.png 336w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-66-300x171.png 300w\" sizes=\"(max-width: 336px) 100vw, 336px\" \/><br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74358\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-67.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 67\" width=\"358\" height=\"612\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-67.png 358w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-67-175x300.png 175w\" sizes=\"(max-width: 358px) 100vw, 358px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74359\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-68.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 68\" width=\"304\" height=\"283\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-68.png 304w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-68-300x279.png 300w\" sizes=\"(max-width: 304px) 100vw, 304px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74360\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-69.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 69\" width=\"388\" height=\"537\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-69.png 388w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-69-217x300.png 217w\" sizes=\"(max-width: 388px) 100vw, 388px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74361\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-70.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 70\" width=\"390\" height=\"286\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-70.png 390w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-70-300x220.png 300w\" sizes=\"(max-width: 390px) 100vw, 390px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74362\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-71.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 71\" width=\"367\" height=\"569\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-71.png 367w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-71-193x300.png 193w\" sizes=\"(max-width: 367px) 100vw, 367px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74363\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-72.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 72\" width=\"385\" height=\"573\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-72.png 385w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-72-202x300.png 202w\" sizes=\"(max-width: 385px) 100vw, 385px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74364\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-73.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 73\" width=\"327\" height=\"397\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-73.png 327w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-73-247x300.png 247w\" sizes=\"(max-width: 327px) 100vw, 327px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74365\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-74.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 74\" width=\"357\" height=\"455\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-74.png 357w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-74-235x300.png 235w\" sizes=\"(max-width: 357px) 100vw, 357px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 5.<br \/>\nFind the equations of the hyperbola satisfying the given conditions.<br \/>\n(i) Vertices (\u00b12,0, foci = (\u00b1 3,0)<br \/>\n(ii) Vertices = (0, \u00b1 5), foci = (0, \u00b1 8)<br \/>\n(iii) Vertices = (0, \u00b1 3), foci =(0, \u00b1 5)<br \/>\n(iv) Foci = (\u00b1 5,0) length of transverse axis = 8<br \/>\n(v) Foci = (0, \u00b113), length of conjugate axis = 24.<br \/>\n(vi) Foci = \\((\\pm 3 \\sqrt{5}, 0) \\), length of latus rectum = 12<br \/>\n(vii) Foci = (\u00b1 4,0) length of latus rectum -12<br \/>\n(viii) Foci =\\((\\pm 7,0), e=\\frac{4}{3}\\)<br \/>\n(ix) Foci = \\((0, \\pm \\sqrt{10}) \\)passing through (2,3)<br \/>\n(x) Foci = (0, \u00b13) vertices \\(=\\left(0, \\pm \\frac{\\sqrt{11}}{2}\\right)\\)<br \/>\n(xi) Foci = (0, \u00b1 12), length of latus rectum = 36.<br \/>\nAnswer:<br \/>\n(i) Given, Vertices (\u00b12,0, foci =(\u00b13,0)<br \/>\nSince foci lie on the x-axis then the equation of the hyperbola is of the form<br \/>\n\\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1 \\ldots(1)\\)<br \/>\nWe have,<br \/>\nvertices = (\u00b1a, 0) = (\u00b12, 0) \u2234 a =2<br \/>\nAlso foci = (\u00b1 c,0) = (\u00b13,0)\u2234 c = 3<br \/>\nSince c<sup>2<\/sup> = a<sup>2<\/sup> +b<sup>2<br \/>\n<\/sup>\u2234 b<sup>2<\/sup> =c<sup>2<\/sup> &#8211; a<sup>2<\/sup> = (3)<sup>2<\/sup> &#8211; (2)<sup>2<\/sup> =9-4 = 5<br \/>\n\u2234 (1) \\(\\Rightarrow \\frac{x^{2}}{4}-\\frac{y^{2}}{5}=1\\)<\/p>\n<p>(ii) Given Vertices = (0, \u00b1 5), foci = (0, \u00b1 8)<br \/>\nSince foci lie on the y-axis then the required equation of the hyperbola is of the form<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74366\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-75.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 75\" width=\"315\" height=\"197\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-75.png 315w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-75-300x188.png 300w\" sizes=\"(max-width: 315px) 100vw, 315px\" \/><\/p>\n<p>(iii) \\(\\frac{y^{2}}{9}-\\frac{x^{2}}{16}=1\\) (Try yourself)<\/p>\n<p>(iv) Given foci = (\u00b1 5, 0) T-axis = 8<br \/>\nSince, foci lie on the x-axis then the required equation of the hyperbola is of the form 2<br \/>\n\\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1 \\) We have,<br \/>\nfoci =(\u00b1c,0) = (\u00b15,0) \u2234 c = 5<br \/>\nLength of T-axis = 2a = 8 \u2234 a = 4<br \/>\nSince b<sup>2<\/sup>=c<sup>2<\/sup>-a<sup>2<\/sup> =25-16 = 9<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74367\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-76.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 76\" width=\"160\" height=\"54\" \/><\/p>\n<p>(v) Given foci = (0, \u00b113), c-axis length = 24.<br \/>\nSince foci lie on the y-axis then the equation of the hyperbola is of the form<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74368\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-77.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 77\" width=\"309\" height=\"248\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-77.png 309w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-77-300x241.png 300w\" sizes=\"(max-width: 309px) 100vw, 309px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>(vi) Given foci \\(=(\\pm 3 \\sqrt{5}, 0) \\) length of LR = 8<br \/>\nSince foci lie on the x-axis then the required equation of the hyperbola is of the form<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74369\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-78.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 78\" width=\"395\" height=\"608\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-78.png 395w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-78-195x300.png 195w\" sizes=\"(max-width: 395px) 100vw, 395px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74370\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-79.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 79\" width=\"355\" height=\"472\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-79.png 355w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-79-226x300.png 226w\" sizes=\"(max-width: 355px) 100vw, 355px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74371\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-80.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 80\" width=\"418\" height=\"599\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-80.png 418w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-80-209x300.png 209w\" sizes=\"(max-width: 418px) 100vw, 418px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74372\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-81.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 81\" width=\"351\" height=\"451\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-81.png 351w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-81-233x300.png 233w\" sizes=\"(max-width: 351px) 100vw, 351px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74373\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-82.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 82\" width=\"391\" height=\"355\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-82.png 391w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-82-300x272.png 300w\" sizes=\"(max-width: 391px) 100vw, 391px\" \/><\/p>\n<p><strong>Miscellaneous examples<\/strong><\/p>\n<p>Question 1.<br \/>\nIf a parabolic reflector is 20 cm in diameter and 5 cm deep find the focus.<br \/>\nAnswer:<br \/>\nGiven: AB = 20 cm, OC =5 cm<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74374\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-83.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 83\" width=\"194\" height=\"148\" \/><br \/>\n\u2234 AC &#8211; BC = 10 cm (curve is symmetric)<br \/>\n\u2234 A = (5,10), B = (5, &#8211; 10), C = (5, 0)<br \/>\nLet the equation of the parabola be<br \/>\ny<sup>2<\/sup> =4ax<br \/>\nSince A (5, 10) lies on (1), then we have,<br \/>\n(10)<sup>2<\/sup> = 4a(5)<br \/>\n\u2234100 = 20 a \u2234 a =5<br \/>\n\u2234 Focus = (a, 0) = (5, 0) = c, the mid point of the diameter.<br \/>\ni.e., Focus of the reflector is at the mid point of the given diameter.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 2.<br \/>\nAn arch is in the form of a parabola with its axis vertical. The arch is 10 m in high and 5 m wide at the base. How wide is it 2m from the vertex of the parabola?<br \/>\nAnswer:<br \/>\nGiven that AB = 5m, OC = 10m OD = 2m<br \/>\nClearly, AC = CB = = \\(\\frac{5}{2} \\)because the curve is symmetric about y-axis.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74376\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-84.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 84\" width=\"342\" height=\"676\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-84.png 342w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-84-152x300.png 152w\" sizes=\"(max-width: 342px) 100vw, 342px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74377\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-85.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 85\" width=\"390\" height=\"243\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-85.png 390w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-85-300x187.png 300w\" sizes=\"(max-width: 390px) 100vw, 390px\" \/><\/p>\n<p>Question 3.<br \/>\nThe cable of a uniformly loaded suspension bridge hands in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle.<br \/>\nAnswer:<br \/>\nCable is in the form of an upward parabola, its equation is<br \/>\nx<sup>2 <\/sup>= 4 ay &#8230;&#8230;&#8230;&#8230;&#8230;&#8230;&#8230;&#8230;. (1)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74378\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-86.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 86\" width=\"399\" height=\"625\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-86.png 399w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-86-192x300.png 192w\" sizes=\"(max-width: 399px) 100vw, 399px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74379\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-87.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 87\" width=\"404\" height=\"301\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-87.png 404w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-87-300x224.png 300w\" sizes=\"(max-width: 404px) 100vw, 404px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 4.<br \/>\nAn arch is in the form of a semi ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end.<br \/>\nAnswer:<br \/>\nEquation of the arch be<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74380\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-88.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 88\" width=\"392\" height=\"688\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-88.png 392w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-88-171x300.png 171w\" sizes=\"(max-width: 392px) 100vw, 392px\" \/><br \/>\nRequired height of the arch at a point Q, 1.5 m from one end A is 1.56 m.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 5.<br \/>\nA rod of length 12 cm moves with its ends always touching the co-ordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.<br \/>\nAnswer:<br \/>\nLet AB = 12 cm be the rod at any position meet x-axis at A (a, 0) and y-axis at B (0, b).<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74381\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-89.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 89\" width=\"294\" height=\"285\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74382\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-90.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 90\" width=\"389\" height=\"584\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-90.png 389w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-90-200x300.png 200w\" sizes=\"(max-width: 389px) 100vw, 389px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74383\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-91.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 91\" width=\"397\" height=\"223\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-91.png 397w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-91-300x169.png 300w\" sizes=\"(max-width: 397px) 100vw, 397px\" \/><\/p>\n<p>Question 6.<br \/>\nFind the area of the triangle formed by the line joining the vertex of the parabola jc<sup>2<\/sup> = 12y to the ends of its latus rectum.<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74384\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-92.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 92\" width=\"344\" height=\"633\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-92.png 344w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-92-163x300.png 163w\" sizes=\"(max-width: 344px) 100vw, 344px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 7.<br \/>\nA man running a race source notes that the stun of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.<br \/>\nAnswer:<br \/>\nRequired equation is of the form<br \/>\n\\(\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74385\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-93.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 93\" width=\"407\" height=\"416\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-93.png 407w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-93-294x300.png 294w\" sizes=\"(max-width: 407px) 100vw, 407px\" \/><\/p>\n<p>Question 8.<br \/>\nAn equilater triangle is inscribed in the parabola y<sup>2<\/sup> = 4ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.<br \/>\nAnswer:<br \/>\nGiven: \u0394AOB is an equilateral triangle inscribed in the parabola y<sup>2<\/sup> = 4ax,<br \/>\nwhose vertex is &#8216;O&#8217;<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74386\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-94.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 94\" width=\"390\" height=\"534\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-94.png 390w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-94-219x300.png 219w\" sizes=\"(max-width: 390px) 100vw, 390px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74387\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-95.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 95\" width=\"352\" height=\"429\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-95.png 352w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-95-246x300.png 246w\" sizes=\"(max-width: 352px) 100vw, 352px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 9.<br \/>\nThe focus of a parabolic mirror as shown in the figure is at a distance of 5 cm from its vertex. If the mirror is 45 cm deep, find the distance AB.<br \/>\nAnswer:<br \/>\nFrom the given figure, we have equation of the parabolic mirror is<br \/>\ny<sup>2<\/sup> = 4ax &#8230;&#8230;&#8230;&#8230;&#8230;..(1)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74388\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-96.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 96\" width=\"273\" height=\"212\" \/><br \/>\nGiven: OF =5, F be the focus of (1)<br \/>\nSince F = (a, 0) = (5,0)\u00a0 \u2234\u00a0 a = 5<br \/>\nAnd OC = 45 = Depth of the mirror.<br \/>\nA = (45,y) and B = (45,-y)<br \/>\nWe have, x = 45, y = y and a = 5 then equation (1) gives,<br \/>\ny<sup>1<\/sup> =4 x 5 x 45 = 900 y = \u00b1 30<br \/>\nAB = 2y = 2 x 30 = 60 cm<\/p>\n<p>Question 10.<br \/>\nA rod AB, 15 cm rests in between two co-ordinate axes such that A lies on x-axis and B lies on y-axis. A point P (or, y) is a point on AB such t hat AP = 6 Show that the locus of P is an ellipse.<br \/>\nAnswer:<br \/>\nGiven: AB = 15, AP = 6 and PB = 9<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-74389\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-97.png\" alt=\"1st PUC Maths Question Bank Chapter 11 Conic Sections 97\" width=\"362\" height=\"617\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-97.png 362w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-11-Conic-Sections-97-176x300.png 176w\" sizes=\"(max-width: 362px) 100vw, 362px\" \/><\/p>\n<p><strong>Validating statements :<\/strong><\/p>\n<p>Validity of a statement (Truthfulness of a statement) can be established by the following methods.<\/p>\n<ul>\n<li>Direct method<\/li>\n<li>Contradiction method<\/li>\n<li>Contrapositive method<\/li>\n<li>Counter example method<\/li>\n<\/ul>\n<p>Note: In mathematics, counter examples are used to disprove the statements.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 1.<br \/>\nShow that the statement p: \u201cIf x is a real number such that x<sup>3<\/sup> + 4x = 0, then x is O\u201d is true by<br \/>\n(i) Direct method,<br \/>\n(ii) Method of contradiction<br \/>\n(iii) Method of contrapositive.<br \/>\nAnswer:<br \/>\nGiven that, p: \u201cIf x is a real number such that x + 4x = 0, then x is O\u201d.<br \/>\n(i) Direct method:<br \/>\nLet x<sup>3<\/sup>+4x = 0, x \u2208 R<br \/>\n\u21d2 x(x<sup>2<\/sup> + 4) = 0, x \u2208 R<br \/>\n\u21d2 x = 0 \u2235 x<sup>2 <\/sup>+ 4 = 0<br \/>\n\u21d2 \\(x=\\pm \\sqrt{-4} \\notin R\\)<br \/>\n\u21d2 p is a true statement.<\/p>\n<p>(ii) Method of contradiction:<br \/>\nLet x \u2260 O, x \u2208 R<br \/>\ni.e., Let x be a nonzero real number<br \/>\n\u21d2\u00a0x<sup>2<\/sup> &gt; 0 (y square of non-zero real number is always positive)<br \/>\n\u21d2 x<sup>2 <\/sup>+ 4 &gt; 0 + 4<br \/>\n\u21d2 x<sup>2<\/sup>+ 4 &gt; 4 x<sup>2<\/sup> + 4 \u2260 0<br \/>\n\u21d2 x(x<sup>2<\/sup> +4) \u2260 0<br \/>\n\u21d2 x<sup>3<\/sup> + 4x = 0, which is a contradiction for x<sup>3<\/sup> + 4x = 0 .<br \/>\nHence x = 0 \u21d2 p is true.<\/p>\n<p>(iii) Method of contrapositive<br \/>\n\u2235 p: \u201cIf x is a real number such that x<sup>3<\/sup> + 4x = 0, then x is O\u201d.<br \/>\nLet q : \u201cx is a real number and x<sup>3<\/sup> + 4x = 0&#8243; r:&#8221;x is 0&#8243;<br \/>\n\u2234\u00a0 p is q \u21d2 r<br \/>\nIts contrapositive is ~ r \u21d2 ~ q<br \/>\ni.e., \u201cI-x is non-zero real number, then x<sup>3<\/sup> + 4x is also non-zero\u201d.<br \/>\nNow, x \u2208 R x 0<br \/>\n\u21d2 x<sup>2<\/sup> &gt;0<br \/>\n\u21d2 x<sup>2<\/sup>+4&gt;0 + 4<br \/>\n\u21d2 x<sup>2 <\/sup>+ 4 &gt; 4<br \/>\n\u21d2 x<sup>2 <\/sup>+ 4 \u2260 0<br \/>\n\u21d2 x(x<sup>2<\/sup>+4)\u2260 0<br \/>\n\u21d2 x<sup>3<\/sup> + 4x \u2260 0<br \/>\ni.e., ~ r \u21d2~ q is always true.<br \/>\nHence q \u21d2 r is true, i.e., p is true.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 2.<br \/>\nShow that the statement \u201cFor any real numbers a and b, a<sup>2<\/sup> = b<sup>2<\/sup> implies that a = b\u201d is not true by giving counter example.<br \/>\nAnswer:<br \/>\nLet a = 2 and b = &#8211; 2, then a and b are real numbers and a<sup>2<\/sup> = b<sup>2<br \/>\n<\/sup>\u2235 a<sup>2<\/sup> = (2)<sup>2<\/sup> = 4, b<sup>2<\/sup> = (-2)<sup>2<\/sup> = 4<br \/>\na<sup>2<\/sup> = b<sup>2<\/sup> but a \u2260 b (\u2235 2 \u2260 &#8211; 2)<br \/>\nHence, a,b \u2208 R and a<sup>2<\/sup> = b<sup>2<br \/>\n<\/sup>\u21d2 a = b is not true.<\/p>\n<p>Question 3.<br \/>\nShow that the following statement is true by the method of contrapositive, p : If x is an integer and x<sup>2<\/sup> is even, then x is also even.<br \/>\nAnswer:<br \/>\nLet q : x \u2208 I and x<sup>2<\/sup> is even<br \/>\nr: x is even integer.<br \/>\n\u2234 Given p is q \u21d2 r<br \/>\nIts contrapositive is ~ r\u00a0 \u21d2\u00a0 ~ q.<br \/>\nNow, ~ r: x is not an even integer i.e., ~ r<br \/>\n\u21d2 x is an odd integer.<br \/>\n\u21d2x = 2n + 1 for n \u2208 1<br \/>\n\u21d2 x<sup>2<\/sup> = (2n +1)<sup>2 <\/sup>= 4n<sup>2<\/sup> + 4n +1 = 2(2n<sup>2<\/sup> + 2n) +1<br \/>\n\u21d2 x<sup>2<\/sup> is an odd integer.<br \/>\n\u21d2\u00a0 x<sup>2<\/sup> is not an even integer.<br \/>\n\u21d2 ~q<br \/>\n\u2234 ~ r \u21d2 ~ q is true.<br \/>\nHence q \u21d2 r is true, i.e., p is true.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 4.<br \/>\nBy giving counter example, show that the following statements are not true.<br \/>\n(i) If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.(ii) The equation x<sup>2<\/sup> &#8211; 1 = 0 does not have a root lying between 0 and 2.<br \/>\nAnswer:<br \/>\n(i) Let PQR be triangle in which P = 60\u00b0, Q = 60\u00b0, R = 60\u00b0, then the triangle PQR is not an obtuse angled triangle. \u2234\u00a0 Given statement is not true.<br \/>\n(ii) Clearly x<sup>2<\/sup> -1 = 0 has x = 1 as one of the root lies between 0 and 2.<br \/>\n\u2234 Given statement is not true.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Students can Download Maths Chapter 11 Conic Sections Questions and Answers, Notes Pdf, 1st PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations. Karnataka 1st PUC Maths Question Bank Chapter 11 Conic Sections Question 1. Name the different conics. Answer: Circle, &hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[81],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/15124"}],"collection":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/comments?post=15124"}],"version-history":[{"count":0,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/15124\/revisions"}],"wp:attachment":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/media?parent=15124"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/categories?post=15124"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/tags?post=15124"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}