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_(1)` and `e_(2)` are the eccentricities of the ellipse and the hyperbola, respectively, then prove that `(1)/(e_(1)^(2))+(1)/(e_(2)^(2))=2`.

Explanation

let focii of eclipse`= (+-ae,0)`

let `e_1` be eccentricity of ellipse

let`e_2` be the eccentricity of hyperbola

hyperbola focii`= (+-Ae_2,0)`

`ae_1=Ae_2 `

`a=A`

`e_1=e_2`

we have to find `1/e_1^2 + 1/e_2^2 = 1/e^2+1/e^2 `

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