If the eccentricity of a hyperbola is √2 and if the distance between th

Question

If the eccentricity of a hyperbola is $\sqrt{2}$ and if the distance between the foci is 16, then its equation is

Options:

A .

x^{2} - y^{2} = 4

B .

x^{2} - y^{2} = 8

C .

x^{2} - y^{2} = 16

D .

x^{2} - y^{2} = 32

Answer: Option D
:
D

Given that distance between foci, 2c =16
$c = 8; e = \sqrt{2}$
$a = \frac{c}{e} = \frac{8}{\sqrt{2}} = 4 \sqrt{2} b = \sqrt{c^{2} - a^{2}} = \sqrt{32} = 4 \sqrt{2}$

The equation of the hyperbola is
$\frac{x^{2}}{32} - \frac{y^{2}}{32} = 1 \Rightarrow x^{2} - y^{2} = 32$

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