JEE Exam > JEE Notes > Mathematics (Maths) for JEE Main & Advanced > Solved Examples: Ellipse
Solved Examples: Ellipse | Mathematics (Maths) for JEE Main & Advanced PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1 Solved Examples on Ellipse Q1: The length of the latus-rectum of the ellipse ?? ?? ?? + ?? ?? ?? = ???? is (a) v ?? /?? (b) v ?? /?? (c) ?? /?? (d) ???? /?? Ans: (d) Sol: Here the ellipse is ?? 2 9 + ?? 2 5 = 1 Here ?? 2 = 9 and ?? 2 = 5. So, latus-rectum = 2 ?? 2 ?? = 2 ( 5 ) 3 = 10 3 . Q2: What is the equation of the ellipse with foci ( ± ?? , ?? ) and eccentricity ?? ?? (a) ?? ?? ?? + ?? ?? ?? = ???? (b) ?? ?? ?? + ?? ?? ?? = ???? (c) ?? ?? ?? + ?? ?? ?? = ?? (d) ?? ?? ?? + ?? ?? ?? = ?? Ans: (a) Sol: Here ???? = ± 2 , ? ?? = 1 2 , ? ?? = ± 4 Form ?? 2 = ?? 2 ( 1 - ?? 2 ) ? ?? 2 = 16 ( 1 - 1 4 ) ? ?? 2 = 12 Hence, the equation of ellipse is ?? 2 16 + ?? 2 12 = 1 or 3 ?? 2 + 4 ?? 2 = 48 Q3: The eccentricity of an ellipse, with its centre at the origin is ?? ?? . If one of the directrices is ?? = ?? , then the equation of the ellipse is (a) ?? ?? ?? + ?? ?? ?? = ?? (b) ?? ?? ?? + ?? ?? ?? = ???? (c) ?? ?? ?? + ?? ?? ?? = ???? (d) ?? ?? ?? + ?? ?? ?? = ?? Ans: (b) Sol: Given ?? = 1 2 , ?? ?? = 4. So, ?? = 2 ? ?? 2 = 4 Page 2 Solved Examples on Ellipse Q1: The length of the latus-rectum of the ellipse ?? ?? ?? + ?? ?? ?? = ???? is (a) v ?? /?? (b) v ?? /?? (c) ?? /?? (d) ???? /?? Ans: (d) Sol: Here the ellipse is ?? 2 9 + ?? 2 5 = 1 Here ?? 2 = 9 and ?? 2 = 5. So, latus-rectum = 2 ?? 2 ?? = 2 ( 5 ) 3 = 10 3 . Q2: What is the equation of the ellipse with foci ( ± ?? , ?? ) and eccentricity ?? ?? (a) ?? ?? ?? + ?? ?? ?? = ???? (b) ?? ?? ?? + ?? ?? ?? = ???? (c) ?? ?? ?? + ?? ?? ?? = ?? (d) ?? ?? ?? + ?? ?? ?? = ?? Ans: (a) Sol: Here ???? = ± 2 , ? ?? = 1 2 , ? ?? = ± 4 Form ?? 2 = ?? 2 ( 1 - ?? 2 ) ? ?? 2 = 16 ( 1 - 1 4 ) ? ?? 2 = 12 Hence, the equation of ellipse is ?? 2 16 + ?? 2 12 = 1 or 3 ?? 2 + 4 ?? 2 = 48 Q3: The eccentricity of an ellipse, with its centre at the origin is ?? ?? . If one of the directrices is ?? = ?? , then the equation of the ellipse is (a) ?? ?? ?? + ?? ?? ?? = ?? (b) ?? ?? ?? + ?? ?? ?? = ???? (c) ?? ?? ?? + ?? ?? ?? = ???? (d) ?? ?? ?? + ?? ?? ?? = ?? Ans: (b) Sol: Given ?? = 1 2 , ?? ?? = 4. So, ?? = 2 ? ?? 2 = 4 From ?? 2 = ?? 2 ( 1 - ?? 2 ) ? ?? 2 = 4 ( 1 - 1 4 ) = 4 × 3 4 = 3 Hence the equation of ellipse is ?? 2 4 + ?? 2 3 = 1, i.e. 3 ?? 2 + 4 ?? 2 = 12 Q4: The curve represented by ?? = ?? ( ?? ?? ?? ? ?? + ?? ???? ? ?? ) , ?? = ?? ( ?? ?? ?? ? ?? - ?? ???? ? ?? ) is (a) Ellipse (b) Parabola (c) Hyperbola (d) Circle Ans: (a) Sol: Given, ?? = 3 ( cos ? ?? + sin ? ?? ) , ?? = 4 ( cos ? ?? - sin ? ?? ) ? ?? 3 = ( cos ? ?? + sin ? ?? ) , ?? 4 = ( cos ? ?? - sin ? ?? ) Squaring and adding, we get ?? 2 9 + ?? 2 16 = ( 1 + sin ? 2 ?? ) + ( 1 - sin ? 2 ?? ) ? ?? 2 9 + ?? 2 16 = 2, which represents ellipse. Q5: The foci of the ellipse ???? ( ?? + ?? ) ?? + ?? ( ?? + ?? ) ?? = ?????? are (a) ( - ?? , ?? ) , ( ?? , ?? ) (b) ( - ?? , - ?? ) , ( ?? , ?? ) (c) ( ?? , - ?? ) , ( ?? , - ?? ) (d) ( - ?? , ?? ) , ( - ?? , - ?? ) Ans: (d) Sol: Given ellipse is ( ?? + 1 ) 2 9 + ( ?? + 2 ) 2 25 = 1 i.e. ?? 2 9 + ?? 2 25 = 1, where ?? = ?? + 1 and ?? = ?? + 2 Here ?? 2 = 25 , ?? 2 = 9 [Type : ?? 2 ?? 2 + ?? 2 ?? 2 = 1 ] Eccentricity is given by ?? 2 = ?? 2 - ?? 2 ?? 2 = 25 - 9 25 = 16 25 , ? ?? = 4 5 Foci are given by ?? = ± ???? = ± 5 ( 4 5 ) = ± 4 ?? = 0 ? ?? + 2 = ± 4 ? ?? = - 2 ± 4 = - 6 or 2 ?? + 1 = 0 ? ?? = - 1. Hence foci are ( - 1 , - 6 ) or ( - 1 , 2 ). Page 3 Solved Examples on Ellipse Q1: The length of the latus-rectum of the ellipse ?? ?? ?? + ?? ?? ?? = ???? is (a) v ?? /?? (b) v ?? /?? (c) ?? /?? (d) ???? /?? Ans: (d) Sol: Here the ellipse is ?? 2 9 + ?? 2 5 = 1 Here ?? 2 = 9 and ?? 2 = 5. So, latus-rectum = 2 ?? 2 ?? = 2 ( 5 ) 3 = 10 3 . Q2: What is the equation of the ellipse with foci ( ± ?? , ?? ) and eccentricity ?? ?? (a) ?? ?? ?? + ?? ?? ?? = ???? (b) ?? ?? ?? + ?? ?? ?? = ???? (c) ?? ?? ?? + ?? ?? ?? = ?? (d) ?? ?? ?? + ?? ?? ?? = ?? Ans: (a) Sol: Here ???? = ± 2 , ? ?? = 1 2 , ? ?? = ± 4 Form ?? 2 = ?? 2 ( 1 - ?? 2 ) ? ?? 2 = 16 ( 1 - 1 4 ) ? ?? 2 = 12 Hence, the equation of ellipse is ?? 2 16 + ?? 2 12 = 1 or 3 ?? 2 + 4 ?? 2 = 48 Q3: The eccentricity of an ellipse, with its centre at the origin is ?? ?? . If one of the directrices is ?? = ?? , then the equation of the ellipse is (a) ?? ?? ?? + ?? ?? ?? = ?? (b) ?? ?? ?? + ?? ?? ?? = ???? (c) ?? ?? ?? + ?? ?? ?? = ???? (d) ?? ?? ?? + ?? ?? ?? = ?? Ans: (b) Sol: Given ?? = 1 2 , ?? ?? = 4. So, ?? = 2 ? ?? 2 = 4 From ?? 2 = ?? 2 ( 1 - ?? 2 ) ? ?? 2 = 4 ( 1 - 1 4 ) = 4 × 3 4 = 3 Hence the equation of ellipse is ?? 2 4 + ?? 2 3 = 1, i.e. 3 ?? 2 + 4 ?? 2 = 12 Q4: The curve represented by ?? = ?? ( ?? ?? ?? ? ?? + ?? ???? ? ?? ) , ?? = ?? ( ?? ?? ?? ? ?? - ?? ???? ? ?? ) is (a) Ellipse (b) Parabola (c) Hyperbola (d) Circle Ans: (a) Sol: Given, ?? = 3 ( cos ? ?? + sin ? ?? ) , ?? = 4 ( cos ? ?? - sin ? ?? ) ? ?? 3 = ( cos ? ?? + sin ? ?? ) , ?? 4 = ( cos ? ?? - sin ? ?? ) Squaring and adding, we get ?? 2 9 + ?? 2 16 = ( 1 + sin ? 2 ?? ) + ( 1 - sin ? 2 ?? ) ? ?? 2 9 + ?? 2 16 = 2, which represents ellipse. Q5: The foci of the ellipse ???? ( ?? + ?? ) ?? + ?? ( ?? + ?? ) ?? = ?????? are (a) ( - ?? , ?? ) , ( ?? , ?? ) (b) ( - ?? , - ?? ) , ( ?? , ?? ) (c) ( ?? , - ?? ) , ( ?? , - ?? ) (d) ( - ?? , ?? ) , ( - ?? , - ?? ) Ans: (d) Sol: Given ellipse is ( ?? + 1 ) 2 9 + ( ?? + 2 ) 2 25 = 1 i.e. ?? 2 9 + ?? 2 25 = 1, where ?? = ?? + 1 and ?? = ?? + 2 Here ?? 2 = 25 , ?? 2 = 9 [Type : ?? 2 ?? 2 + ?? 2 ?? 2 = 1 ] Eccentricity is given by ?? 2 = ?? 2 - ?? 2 ?? 2 = 25 - 9 25 = 16 25 , ? ?? = 4 5 Foci are given by ?? = ± ???? = ± 5 ( 4 5 ) = ± 4 ?? = 0 ? ?? + 2 = ± 4 ? ?? = - 2 ± 4 = - 6 or 2 ?? + 1 = 0 ? ?? = - 1. Hence foci are ( - 1 , - 6 ) or ( - 1 , 2 ). Q6: If any tangent to the ellipse ?? ?? ?? ?? + ?? ?? ?? ?? = ?? intercepts equal lengths ?? on the axes, then ?? = (a) ?? ?? + ?? ?? (b) v ?? ?? + ?? ?? (c) ( ?? ?? + ?? ?? ) ?? (d) None of these Ans: (b) Sol: The equation of any tangent to the given ellipse is ?? ?? cos ? ?? + ?? ?? sin ? ?? = 1 This line meets the coordinate axes at ?? ( ?? cos ? ?? , 0 ) and ?? (0 , ?? s in ? ?? ) ? ?? cos ? ?? = ?? = ?? s in ? ?? ? cos ? ?? = ?? ?? and sin ? ?? = ?? ?? ? cos 2 ? ?? + sin 2 ? ?? = ?? 2 ?? 2 + ?? 2 ?? 2 ? ?? 2 = ?? 2 + ?? 2 ? ?? = v ?? 2 + ?? 2 . Q7: The locus of the middle point of the intercept of the tangents drawn from an external point to the ellipse ?? ?? + ?? ?? ?? = ?? between the coordinate axes, is (a) ?? ?? ?? + ?? ?? ?? ?? = ?? (b) ?? ?? ?? ?? + ?? ?? ?? ?? = ?? (c) ?? ?? ?? ?? + ?? ?? ?? ?? = ?? (d) ?? ?? ?? ?? + ?? ?? ?? = ?? Ans: (c) Sol: Let the point of contact be ?? = ( v 2 cos ? ?? , sin ? ?? ) Equation of tangent ???? is ?? v 2 cos ? ?? + ?? sin ? ?? = 1 ? ?? = ( v 2 sec ? ?? , 0 ) ; ?? = ( 0 , coses ? ?? ) Let the middle point ?? of ???? be ( h , ?? ). ? h = sec ? ?? v 2 , ?? = cosec ? ?? 2 ? cos ? ?? = 1 h v 2 , sin ? ?? = 1 2 ?? ? 1 2 h 2 + 1 4 ?? 2 = 1 Page 4 Solved Examples on Ellipse Q1: The length of the latus-rectum of the ellipse ?? ?? ?? + ?? ?? ?? = ???? is (a) v ?? /?? (b) v ?? /?? (c) ?? /?? (d) ???? /?? Ans: (d) Sol: Here the ellipse is ?? 2 9 + ?? 2 5 = 1 Here ?? 2 = 9 and ?? 2 = 5. So, latus-rectum = 2 ?? 2 ?? = 2 ( 5 ) 3 = 10 3 . Q2: What is the equation of the ellipse with foci ( ± ?? , ?? ) and eccentricity ?? ?? (a) ?? ?? ?? + ?? ?? ?? = ???? (b) ?? ?? ?? + ?? ?? ?? = ???? (c) ?? ?? ?? + ?? ?? ?? = ?? (d) ?? ?? ?? + ?? ?? ?? = ?? Ans: (a) Sol: Here ???? = ± 2 , ? ?? = 1 2 , ? ?? = ± 4 Form ?? 2 = ?? 2 ( 1 - ?? 2 ) ? ?? 2 = 16 ( 1 - 1 4 ) ? ?? 2 = 12 Hence, the equation of ellipse is ?? 2 16 + ?? 2 12 = 1 or 3 ?? 2 + 4 ?? 2 = 48 Q3: The eccentricity of an ellipse, with its centre at the origin is ?? ?? . If one of the directrices is ?? = ?? , then the equation of the ellipse is (a) ?? ?? ?? + ?? ?? ?? = ?? (b) ?? ?? ?? + ?? ?? ?? = ???? (c) ?? ?? ?? + ?? ?? ?? = ???? (d) ?? ?? ?? + ?? ?? ?? = ?? Ans: (b) Sol: Given ?? = 1 2 , ?? ?? = 4. So, ?? = 2 ? ?? 2 = 4 From ?? 2 = ?? 2 ( 1 - ?? 2 ) ? ?? 2 = 4 ( 1 - 1 4 ) = 4 × 3 4 = 3 Hence the equation of ellipse is ?? 2 4 + ?? 2 3 = 1, i.e. 3 ?? 2 + 4 ?? 2 = 12 Q4: The curve represented by ?? = ?? ( ?? ?? ?? ? ?? + ?? ???? ? ?? ) , ?? = ?? ( ?? ?? ?? ? ?? - ?? ???? ? ?? ) is (a) Ellipse (b) Parabola (c) Hyperbola (d) Circle Ans: (a) Sol: Given, ?? = 3 ( cos ? ?? + sin ? ?? ) , ?? = 4 ( cos ? ?? - sin ? ?? ) ? ?? 3 = ( cos ? ?? + sin ? ?? ) , ?? 4 = ( cos ? ?? - sin ? ?? ) Squaring and adding, we get ?? 2 9 + ?? 2 16 = ( 1 + sin ? 2 ?? ) + ( 1 - sin ? 2 ?? ) ? ?? 2 9 + ?? 2 16 = 2, which represents ellipse. Q5: The foci of the ellipse ???? ( ?? + ?? ) ?? + ?? ( ?? + ?? ) ?? = ?????? are (a) ( - ?? , ?? ) , ( ?? , ?? ) (b) ( - ?? , - ?? ) , ( ?? , ?? ) (c) ( ?? , - ?? ) , ( ?? , - ?? ) (d) ( - ?? , ?? ) , ( - ?? , - ?? ) Ans: (d) Sol: Given ellipse is ( ?? + 1 ) 2 9 + ( ?? + 2 ) 2 25 = 1 i.e. ?? 2 9 + ?? 2 25 = 1, where ?? = ?? + 1 and ?? = ?? + 2 Here ?? 2 = 25 , ?? 2 = 9 [Type : ?? 2 ?? 2 + ?? 2 ?? 2 = 1 ] Eccentricity is given by ?? 2 = ?? 2 - ?? 2 ?? 2 = 25 - 9 25 = 16 25 , ? ?? = 4 5 Foci are given by ?? = ± ???? = ± 5 ( 4 5 ) = ± 4 ?? = 0 ? ?? + 2 = ± 4 ? ?? = - 2 ± 4 = - 6 or 2 ?? + 1 = 0 ? ?? = - 1. Hence foci are ( - 1 , - 6 ) or ( - 1 , 2 ). Q6: If any tangent to the ellipse ?? ?? ?? ?? + ?? ?? ?? ?? = ?? intercepts equal lengths ?? on the axes, then ?? = (a) ?? ?? + ?? ?? (b) v ?? ?? + ?? ?? (c) ( ?? ?? + ?? ?? ) ?? (d) None of these Ans: (b) Sol: The equation of any tangent to the given ellipse is ?? ?? cos ? ?? + ?? ?? sin ? ?? = 1 This line meets the coordinate axes at ?? ( ?? cos ? ?? , 0 ) and ?? (0 , ?? s in ? ?? ) ? ?? cos ? ?? = ?? = ?? s in ? ?? ? cos ? ?? = ?? ?? and sin ? ?? = ?? ?? ? cos 2 ? ?? + sin 2 ? ?? = ?? 2 ?? 2 + ?? 2 ?? 2 ? ?? 2 = ?? 2 + ?? 2 ? ?? = v ?? 2 + ?? 2 . Q7: The locus of the middle point of the intercept of the tangents drawn from an external point to the ellipse ?? ?? + ?? ?? ?? = ?? between the coordinate axes, is (a) ?? ?? ?? + ?? ?? ?? ?? = ?? (b) ?? ?? ?? ?? + ?? ?? ?? ?? = ?? (c) ?? ?? ?? ?? + ?? ?? ?? ?? = ?? (d) ?? ?? ?? ?? + ?? ?? ?? = ?? Ans: (c) Sol: Let the point of contact be ?? = ( v 2 cos ? ?? , sin ? ?? ) Equation of tangent ???? is ?? v 2 cos ? ?? + ?? sin ? ?? = 1 ? ?? = ( v 2 sec ? ?? , 0 ) ; ?? = ( 0 , coses ? ?? ) Let the middle point ?? of ???? be ( h , ?? ). ? h = sec ? ?? v 2 , ?? = cosec ? ?? 2 ? cos ? ?? = 1 h v 2 , sin ? ?? = 1 2 ?? ? 1 2 h 2 + 1 4 ?? 2 = 1 Thus required locus is 1 2 ?? 2 + 1 4 ?? 2 = 1 Q8: Find the centre, the eccentricity, the foci, the directrices and the lengths and the equations of the axes of the ellipse 5x 2 + 9y 2 + 10x - 36y - 4= 0 Sol: Rewrite the equation in the standard form and compare them to get the centre, eccentricity etc. 5x 2 + 9y 2 + 10x - 36y - 4= 0 , the given equation can be written as 5(x 2 + 2x) + 9(y 2 - 4y) = 4 5(x + 1) 2 + 9(y - 2) 2 = 45 Shift the origin to O ' = ( -1, 2) ? X = x+1; Y = y-2 ……(i) This is in standard form Now for an ellipse in the standard form we have Centre = (0, 0); foci = (±ae, 0); directrices axes x = 0, y = 0, length of major axis = 2a, length of minor axis = 2b. Now for (i) the centre is given by X = 0, Y = 0 ? x + 1= 0, y - 2 = 0 Page 5 Solved Examples on Ellipse Q1: The length of the latus-rectum of the ellipse ?? ?? ?? + ?? ?? ?? = ???? is (a) v ?? /?? (b) v ?? /?? (c) ?? /?? (d) ???? /?? Ans: (d) Sol: Here the ellipse is ?? 2 9 + ?? 2 5 = 1 Here ?? 2 = 9 and ?? 2 = 5. So, latus-rectum = 2 ?? 2 ?? = 2 ( 5 ) 3 = 10 3 . Q2: What is the equation of the ellipse with foci ( ± ?? , ?? ) and eccentricity ?? ?? (a) ?? ?? ?? + ?? ?? ?? = ???? (b) ?? ?? ?? + ?? ?? ?? = ???? (c) ?? ?? ?? + ?? ?? ?? = ?? (d) ?? ?? ?? + ?? ?? ?? = ?? Ans: (a) Sol: Here ???? = ± 2 , ? ?? = 1 2 , ? ?? = ± 4 Form ?? 2 = ?? 2 ( 1 - ?? 2 ) ? ?? 2 = 16 ( 1 - 1 4 ) ? ?? 2 = 12 Hence, the equation of ellipse is ?? 2 16 + ?? 2 12 = 1 or 3 ?? 2 + 4 ?? 2 = 48 Q3: The eccentricity of an ellipse, with its centre at the origin is ?? ?? . If one of the directrices is ?? = ?? , then the equation of the ellipse is (a) ?? ?? ?? + ?? ?? ?? = ?? (b) ?? ?? ?? + ?? ?? ?? = ???? (c) ?? ?? ?? + ?? ?? ?? = ???? (d) ?? ?? ?? + ?? ?? ?? = ?? Ans: (b) Sol: Given ?? = 1 2 , ?? ?? = 4. So, ?? = 2 ? ?? 2 = 4 From ?? 2 = ?? 2 ( 1 - ?? 2 ) ? ?? 2 = 4 ( 1 - 1 4 ) = 4 × 3 4 = 3 Hence the equation of ellipse is ?? 2 4 + ?? 2 3 = 1, i.e. 3 ?? 2 + 4 ?? 2 = 12 Q4: The curve represented by ?? = ?? ( ?? ?? ?? ? ?? + ?? ???? ? ?? ) , ?? = ?? ( ?? ?? ?? ? ?? - ?? ???? ? ?? ) is (a) Ellipse (b) Parabola (c) Hyperbola (d) Circle Ans: (a) Sol: Given, ?? = 3 ( cos ? ?? + sin ? ?? ) , ?? = 4 ( cos ? ?? - sin ? ?? ) ? ?? 3 = ( cos ? ?? + sin ? ?? ) , ?? 4 = ( cos ? ?? - sin ? ?? ) Squaring and adding, we get ?? 2 9 + ?? 2 16 = ( 1 + sin ? 2 ?? ) + ( 1 - sin ? 2 ?? ) ? ?? 2 9 + ?? 2 16 = 2, which represents ellipse. Q5: The foci of the ellipse ???? ( ?? + ?? ) ?? + ?? ( ?? + ?? ) ?? = ?????? are (a) ( - ?? , ?? ) , ( ?? , ?? ) (b) ( - ?? , - ?? ) , ( ?? , ?? ) (c) ( ?? , - ?? ) , ( ?? , - ?? ) (d) ( - ?? , ?? ) , ( - ?? , - ?? ) Ans: (d) Sol: Given ellipse is ( ?? + 1 ) 2 9 + ( ?? + 2 ) 2 25 = 1 i.e. ?? 2 9 + ?? 2 25 = 1, where ?? = ?? + 1 and ?? = ?? + 2 Here ?? 2 = 25 , ?? 2 = 9 [Type : ?? 2 ?? 2 + ?? 2 ?? 2 = 1 ] Eccentricity is given by ?? 2 = ?? 2 - ?? 2 ?? 2 = 25 - 9 25 = 16 25 , ? ?? = 4 5 Foci are given by ?? = ± ???? = ± 5 ( 4 5 ) = ± 4 ?? = 0 ? ?? + 2 = ± 4 ? ?? = - 2 ± 4 = - 6 or 2 ?? + 1 = 0 ? ?? = - 1. Hence foci are ( - 1 , - 6 ) or ( - 1 , 2 ). Q6: If any tangent to the ellipse ?? ?? ?? ?? + ?? ?? ?? ?? = ?? intercepts equal lengths ?? on the axes, then ?? = (a) ?? ?? + ?? ?? (b) v ?? ?? + ?? ?? (c) ( ?? ?? + ?? ?? ) ?? (d) None of these Ans: (b) Sol: The equation of any tangent to the given ellipse is ?? ?? cos ? ?? + ?? ?? sin ? ?? = 1 This line meets the coordinate axes at ?? ( ?? cos ? ?? , 0 ) and ?? (0 , ?? s in ? ?? ) ? ?? cos ? ?? = ?? = ?? s in ? ?? ? cos ? ?? = ?? ?? and sin ? ?? = ?? ?? ? cos 2 ? ?? + sin 2 ? ?? = ?? 2 ?? 2 + ?? 2 ?? 2 ? ?? 2 = ?? 2 + ?? 2 ? ?? = v ?? 2 + ?? 2 . Q7: The locus of the middle point of the intercept of the tangents drawn from an external point to the ellipse ?? ?? + ?? ?? ?? = ?? between the coordinate axes, is (a) ?? ?? ?? + ?? ?? ?? ?? = ?? (b) ?? ?? ?? ?? + ?? ?? ?? ?? = ?? (c) ?? ?? ?? ?? + ?? ?? ?? ?? = ?? (d) ?? ?? ?? ?? + ?? ?? ?? = ?? Ans: (c) Sol: Let the point of contact be ?? = ( v 2 cos ? ?? , sin ? ?? ) Equation of tangent ???? is ?? v 2 cos ? ?? + ?? sin ? ?? = 1 ? ?? = ( v 2 sec ? ?? , 0 ) ; ?? = ( 0 , coses ? ?? ) Let the middle point ?? of ???? be ( h , ?? ). ? h = sec ? ?? v 2 , ?? = cosec ? ?? 2 ? cos ? ?? = 1 h v 2 , sin ? ?? = 1 2 ?? ? 1 2 h 2 + 1 4 ?? 2 = 1 Thus required locus is 1 2 ?? 2 + 1 4 ?? 2 = 1 Q8: Find the centre, the eccentricity, the foci, the directrices and the lengths and the equations of the axes of the ellipse 5x 2 + 9y 2 + 10x - 36y - 4= 0 Sol: Rewrite the equation in the standard form and compare them to get the centre, eccentricity etc. 5x 2 + 9y 2 + 10x - 36y - 4= 0 , the given equation can be written as 5(x 2 + 2x) + 9(y 2 - 4y) = 4 5(x + 1) 2 + 9(y - 2) 2 = 45 Shift the origin to O ' = ( -1, 2) ? X = x+1; Y = y-2 ……(i) This is in standard form Now for an ellipse in the standard form we have Centre = (0, 0); foci = (±ae, 0); directrices axes x = 0, y = 0, length of major axis = 2a, length of minor axis = 2b. Now for (i) the centre is given by X = 0, Y = 0 ? x + 1= 0, y - 2 = 0 Foci are given by X = ±ae, Y= 0 i.e. x +1 = ±2 and y - 2 = 0 i.e. x = 1, y = 2 and x = -3, y= 2 ? Foci = (1, 2); (-3, 2) The equation of directrices are given by The equation of the axes are given by X = 0, Y = 0 i.e. x + 1 = 0, y - 2 = 0 i.e. x = -1, y = 2 Length of the axes being 2a, 2b i.e, 6, 2v5. Q9: A tangent to the ellipse touches it at the point P in the first quadrant and meets the x and y axes in A and B respectively. If P divides AB in the ratio 3 : 1, find the equation of the tangent at P. Sol: Consider a point in the parametric form and obtain the points A and B. Now use the condition that the point P divides AB in the ratio 3:1. Let P = (a cos ?, b sin ?): …….(i) Equation of the tangent at P(?) isRead More
|
209 videos|443 docs|143 tests
|
Related Exams
About this Document
|
Nov 21, 2024 Last updated |
Document Description: Solved Examples: Ellipse for JEE 2024 is part of Mathematics (Maths) for JEE Main & Advanced preparation.
The notes and questions for Solved Examples: Ellipse have been prepared according to the JEE exam syllabus. Information about Solved Examples: Ellipse covers topics
like and Solved Examples: Ellipse Example, for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Solved Examples: Ellipse.
Introduction of Solved Examples: Ellipse in English is available as part of our Mathematics (Maths) for JEE Main & Advanced
for JEE & Solved Examples: Ellipse in Hindi for Mathematics (Maths) for JEE Main & Advanced course.
Download more important topics related with notes, lectures and mock test series for JEE
Exam by signing up for free. JEE: Solved Examples: Ellipse | Mathematics (Maths) for JEE Main & Advanced
Description
Full syllabus notes, lecture & questions for Solved Examples: Ellipse | Mathematics (Maths) for JEE Main & Advanced - JEE | Plus excerises question with solution to help you revise complete syllabus for Mathematics (Maths) for JEE Main & Advanced | Best notes, free PDF download
Information about Solved Examples: Ellipse
In this doc you can find the meaning of Solved Examples: Ellipse defined & explained in the simplest way possible. Besides explaining types of
Solved Examples: Ellipse theory, EduRev gives you an ample number of questions to practice Solved Examples: Ellipse tests, examples and also practice JEE
tests
|
209 videos|443 docs|143 tests
|
Download as PDF
|
Explore Courses for JEE exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
MCQs
,Free
,Previous Year Questions with Solutions
,shortcuts and tricks
,Solved Examples: Ellipse | Mathematics (Maths) for JEE Main & Advanced
,Exam
,Solved Examples: Ellipse | Mathematics (Maths) for JEE Main & Advanced
,Objective type Questions
,Extra Questions
,study material
,Semester Notes
,practice quizzes
,mock tests for examination
,past year papers
,Sample Paper
,Solved Examples: Ellipse | Mathematics (Maths) for JEE Main & Advanced
,Viva Questions
,Summary
,video lectures
,Important questions
,ppt
;
Additional Information about Solved Examples: Ellipse for JEE Preparation
Solved Examples: Ellipse Free PDF Download
The Solved Examples: Ellipse is an invaluable resource that delves deep into the core of the JEE exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Solved Examples: Ellipse now and kickstart your journey towards success in the JEE exam.
Importance of Solved Examples: Ellipse
The importance of Solved Examples: Ellipse cannot be overstated, especially for JEE aspirants.
This document holds the key to success in the JEE exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Solved Examples: Ellipse Notes
Solved Examples: Ellipse Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Solved Examples: Ellipse.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Solved Examples: Ellipse Notes on EduRev are your ultimate resource for success.
Solved Examples: Ellipse JEE Questions
The "Solved Examples: Ellipse JEE Questions" guide is a valuable resource for all aspiring students preparing for the
JEE exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Solved Examples: Ellipse on the App
Students of JEE can study Solved Examples: Ellipse alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Solved Examples: Ellipse,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Solved Examples: Ellipse is prepared as per the latest JEE syllabus.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup on EduRev and stay on top of your study goals
10M+ students crushing their study goals daily