If eccentricities of hyperbolas frac{{{x^2}}}{{{a^2}}} - frac{{{y^2}}}{{{b^2}}} = 1 and frac{{{y^2}}

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10-2. Parabola, Ellipse, Hyperbola

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If the eccentricities of the hyperbolas $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ and $\frac{{{y^2}}}{{{b^2}}} - \frac{{{x^2}}}{{{a^2}}} = 1$ be e and ${e_1}$, then $\frac{1}{{{e^2}}} + \frac{1}{{e_1^2}} = $

A

$1$

B

$2$

C

$3$

D

None of these

Solution

(a) $e = \sqrt {1 + \frac{{{b^2}}}{{{a^2}}}} $

==> ${e^2} = \frac{{{a^2} + {b^2}}}{{{a^2}}}$

==> ${e_1} = \sqrt {1 + \frac{{{a^2}}}{{{b^2}}}} $

==> $e_1^2 = \frac{{{b^2} + {a^2}}}{{{b^2}}}$

==> $\frac{1}{{e_1^2}} + \frac{1}{{{e^2}}} = 1$.

Standard 11

Mathematics

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