`e_1`and `e_2` are respectively the eccentricities of a hyperbola and its conjugate.Prove that `1/e_1^2+1/e_2^2=1`

e_(1) and e_(2) are respectively the eccentricites of a hyperbola and its conjugate. Prove that (1)/(e_1^(2))+(1)/(e_2^2) =1

If e_1 and e_2 be the eccentricities of a hyperbola and its conjugate, show that 1/(e_1^2)+1/(e_2^2)=1 .

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/(e_1^2)+1/(e_2^2)=2 .

If a variable line has its intercepts on the coordinate axes e and e^(prime), where e/2a n d e^(prime)/2 are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle x^2+y^2=r^2, where r=

If e_(1) and e_(2) are the respective eccentricities of the hyperbola 4x^(2) - 9y^(2) = 36 and the ellipse 2x^(2) + 9y^(2) = 36 , then which of the following relation is true?

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

Prove that : 1leint_0^1e^(x^2)dxlee

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

Consider an ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1(a gt b) . A hyperbola has its vertices at the extremities of minor axis of the ellipse and the length of major axis of the ellipse is equal to the distance between the foci of hyperbola. Let e_(1) and e_(2) be the eccentricities of ellipse and hyperbola, respectively. Also, let A_(1) be the area of the quadrilateral fored by joining all the foci and A_(2) be the area of the quadrilateral formed by all the directries. The relation between e_(1) and e_(2) is given by

If e_(1) and e_(2) be the eccentricities of the hyperbolas 9x^(2) - 16y^(2) = 576 and 9x^(2) - 16y^(2) = 144 respectively, then -