The distance between the foci of a hyperbola is 16, and its eccentricity is 2. Its equation is

(a) $x^{2} - y^{2} = 32$
(b) $\frac{x^{2}}{4} - \frac{y^{2}}{9} = 1$
(c) $2 x - 3 y^{2} = 7$
(d) None of these

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Alok Kumar Singh | Contributor-Level 10

a year ago

This is an objective Type Questions as classified in NCERT Exemplar

Sol:

$W e k n o w t h a t Distance$
$\begin{array}{l} b e t w e e n t h e f o c i = 2 a e \\ ∴ 2 a e = 16 \Rightarrow a e = 8 \\ G i v e n t h a t e = \sqrt[]{2} \\ ∴ \sqrt[]{2} a = 8 \Rightarrow a = 4 \sqrt[]{2} \\ N o w b^{2} = a^{2} (e^{2} - 1) \\ \Rightarrow b^{2} = 32 (2 - 1) \Rightarrow b^{2} = 32 \\ S o, t h e e q u a t i o n o f t h e h y p e r b o l a i s \\ \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \Rightarrow \frac{x^{2}}{32} - \frac{y^{2}}{32} = 1 \Rightarrow x^{2} - y^{2} = 32 \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

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