Home
Class 11
MATHS
If e(1)ande(2) be the eccentricities of ...

If `e_(1)ande_(2)` be the eccentricities of the ellipses `(x^(2))/(a^(2))+(4y^(2))/(b^(2))=1and(x^(2))/(a^(2))+(4y^(2))/(b^(2))=1` respectively then prove that `3=4e_(2)^(2)-e_(1)^(2)`.

Text Solution

Verified by Experts

For ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
Its eccentricity is `e_(1)`
`:." "b^(2)=a^(2)(1-e_(1)^(2))rArr(b^(2))/(a^(2))=1-e_(1)^(2)` . . . (1)
For ellipse `(x^(2))/(a^(2))+(4y^(2))/(b^(2))=1`
`rArr" "(x^(2))/(a^(2))+(4y^(2))/(b^(2)//4)=1`
Its eccentricity is `e_(2)`
`:." "(b^(2))/(4)=a^(2)(1-e_(2)^(2))`
`rArr" "(b^(2))/(4a^(2))=1-e_(2)^(2)" "rArr" "(b^(2))/(a^(2))=4-4e_(2)^(2)` . . . (2)
From eqs. (1) and (2), we get
`4-4e_(2)^(2)=1-e_(1)^(2)`
`rArr" "3=4e_(2)^(2)-e_(1)^(2)`.
Promotional Banner

Topper's Solved these Questions

  • CONIC SECTION

    NAGEEN PRAKASHAN|Exercise Miscellaneous Example|3 Videos

  • CONIC SECTION

    NAGEEN PRAKASHAN|Exercise Exercise 11A|38 Videos

  • COMPLEX NUMBERS AND QUADRATIC EQUATION

    NAGEEN PRAKASHAN|Exercise MISCELLANEOUS EXERCISE|20 Videos

  • INTRODUCTION OF THREE DIMENSIONAL GEOMETRY

    NAGEEN PRAKASHAN|Exercise Miscellaneous Exercise|6 Videos

Similar Questions

Explore conceptually related problems

If e' is the eccentricity of the ellipse (x^(2))/(a^(2)) + (y^(2))/(b^(2)) =1 (a gt b) , then

Statement- 1 : If 5//3 is the eccentricity of a hyperbola, then the eccentricity of its conjugate hyperbola is 5//4 . Statement- 2 : If e and e' are the eccentricities of hyperbolas (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 and (x^(2))/(a^(2))-(y^(2))/(b^(2))=-1 respectively, then (1)/(e^(2))+(1)/(e'^(2))=1 .

If e is the eccentricity of the ellipse (x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1 (a lt b ) , then ,

If the eccentricity of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1and(y^(2))/(b^(2))-(x^(2))/(a^(2))=1" are "e_(1)ande_(2) respectively then prove that : (1)/(e_(1)^(2))+(1)/(e_(2)^(2))

If e is eccentricity of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 (where,a lt b), then

If e_(1) and e_(2) are respectively the eccentricities of the ellipse (x^(2))/(18)+(y^(2))/(4)=1 and the hyperbola (x^(2))/(9)-(y^(2))/(4)=1, then write the value of 2e_(1)^(2)+e_(2)^(2)

The eccentricity of the ellipse x^(2)+4y^(2)+8y-2x+1=0 , is

If e_(1) , and e_(2) are respectively the eccentricities of the conics (x^(2))/(25)-(y^(2))/(11)=1 and (x^(2))/(16)+(y^(2))/(7)=1 then e_(1)e_(2) is equal to