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Definition of Hyperbola

If e and ...

Question

If $e$ and $e^{'}$ be the eccentricities of two conics $S = 0$ and $S^{'} = 0$ and if $e^{2} + {e^{'}}^{2} > 2$ , then $S$ and $S^{'}$ can be

hyperbola

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ellipse

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parabola

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none of these

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Solution

The correct option is C hyperbola
For a parabola the eccentricity is $1$ .
$∴ e^{2} + {e^{'}}^{2} = 1 + 1 = 2$
For an ellipse the eccentricity is less than 1
$∴ 0 < e < 1, 0 < e^{'} < 1 \Rightarrow e^{2} + {e^{'}}^{2} < 2$
For a hyperbola the eccentricity is greater than 1. So, the conics can be hyperbolas

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