Home
Class 12
MATHS
An ellipse and a hyperbola are confocal ...

An ellipse and a hyperbola are confocal (have the same focus) and the conjugate axis of the hyperbola is equal to the minor axis of the ellipse. If `e_1a n de_2` are the eccentricities of the ellipse and the hyperbola, respectively, then prove that `1/(e1 2)+1/(e2 2)=2` .

Text Solution

Verified by Experts

Let the foci of the ellipse be `(pmae_(1),0)` and those of the hyperbola be `(pmAe_(2),0)`

According to the question, we have
`ae_(1)=Ae_(2)" (1)"`
Also, it is given that the conjugate axis of the hyperbola is equals to the minor axis of the ellipse. Therefore,
`a^(2)(1-e_(1)^(2))=A^(2)(e_(2)^(2)-1)" (2)"`
From (1) and (2), we have
`(1)/(e_(1)^(2))+(1)/(e_(2)^(2))=2`
Promotional Banner

Topper's Solved these Questions

  • HYPERBOLA

    CENGAGE|Exercise Exercise 7.3|10 Videos

  • HYPERBOLA

    CENGAGE|Exercise Exercise 7.4|5 Videos

  • HYPERBOLA

    CENGAGE|Exercise Exercise 7.1|3 Videos

  • HIGHT AND DISTANCE

    CENGAGE|Exercise JEE Previous Year|3 Videos

  • INDEFINITE INTEGRATION

    CENGAGE|Exercise Question Bank|21 Videos

Similar Questions

Explore conceptually related problems

Consider an ellipse x^2/a^2+y^2/b^2=1 Let a hyperbola is having its vertices at the extremities of minor axis of an ellipse and length of major axis of an ellipse is equal to the distance between the foci of hyperbola. Let e_1 and e_2 be the eccentricities of an ellipse and hyperbola respectively. Again let A be the area of the quadrilateral formed by joining all the foci and A, be the area of the quadrilateral formed by all the directrices. The relation between e_1 and e_2 is given by

If radii of director circles of x^2/a^2+y^2/b^2=1 and x^2/a^2-y^2/b^2=1 are 2r and r respectively, let e_E and e_H are the eccentricities of ellipse and hyperbola respectively, then

if vertices of an ellipse are (-4,1),(6,1) and x-2y=2 is focal chord then the eccentricity of the ellipse is

If the ellipse x^(2)+2y^(2)=4 and the hyperbola S = 0 have same end points of the latus rectum, then the eccentricity of the hyperbola can be

S and T are the foci of the ellipse x^(2)/a^(2)+y^(2)/b^(2)=1 and B is an end of the minor axis. If STB is an equilateral triangle, the eccentricity of the ellipse is . . .

The foci of an ellipse are (+- 2,0) and its eccentricity is 1/2 then the equation is

If e_1 is the eccentricity of the ellipse x^2/16+y^2/25=1 and e_2 is the eccentricity of the hyperbola passing through the foci of the ellipse and e_1 e_2=1 , then equation of the hyperbola is

P and Q are the foci of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 and B is an end of the minor axis. If P B Q is an equilateral triangle, then the eccentricity of the ellipse is

If a hyperbola passes through the foci of the ellipse (x^2)/(25)+(y^2)/(16)=1 . Its transverse and conjugate axes coincide respectively with the major and minor axes of the ellipse and if the product of eccentricities of hyperbola and ellipse is 1 then the equation of hyperbola is (x^2)/9-(y^2)/(16)=1 b. the equation of hyperbola is (x^2)/9-(y^2)/(25)=1 c. focus of hyperbola is (5, 0) d. focus of hyperbola is (5sqrt(3),0)