Question

The distance between the foci of a hyperbola is 16 and its eccentricity is $\sqrt{2}$. Its equation is

• A. $x^2 - y^2 = 32$
• B. $\frac{x^2}{4} - \frac{y^2}{9} = 1$
• C. $2x^2 - 3y^2 = 7$
• D. $y^2 - x^2 = 32$

Answer 1

Answer: (A)

$x^2 - y^2 = 32$

Answer 2

Given:
Given $2c = 16$
$\Rightarrow\, c = 8$
Eccentricity, $e = \sqrt{2} \, \Rightarrow \, \frac{c}{a} = \sqrt{2}$
$\Rightarrow a = \frac{c}{\sqrt{2}} = \frac{8}{\sqrt{2}}$
We have $b^2 = c^2 - a^2 = 64 - 32 = 32$
$\therefore$ Equation to hyperbola is $\frac{x^2}{32} - \frac{y^2}{32} = 1$
$\Rightarrow x^2 - y^2 = 32$