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` .

Explanation

To prove that \( \frac{1}{e_1^2} + \frac{1}{e_2^2} = 2 \) for a confocal ellipse and hyperbola, we will follow these steps:

### Step 1: Understand the definitions and properties
Let the semi-major axis of the ellipse be \( a_1 \) and the semi-minor axis be \( b_1 \). The eccentricity \( e_1 \) of the ellipse is given by:
\[ e_1 = \sqrt{1 - \frac{b_1^2}{a_1^2}} \]

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