If $e_{1}$ and $e_{2}$ are the eccentricities of two conics with $e_{1}^{2} + e_{2}^{2} = 3$ , then the conics are.

Ellipses

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Parabolas

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Hyperbolas

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Solution

The correct option is D Hyperbolas
Given, $e_{1}^{2} + e_{2}^{2} = 3$
$e_{1}$ and $e_{2}$ are eccentricities of same type of conic.
Let $e_{1} = e_{2}$
$∴ 2 e_{1}^{2} = 3$ (from given equation)
$e_{1}^{2} = \frac{3}{2} = 1.5$
$e_{1} = \sqrt{1.5} > 1$
$∴ e_{1} = e_{2} > 1$
$∴$ Both conics are Hyperbolas. $∵ e > 1$

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