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If the distance between the foci of a hyperbola is 16 and its eccentricity is $\sqrt{2}$, then obtain the equation of the hyperbola.
Solution:
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Verified Answer
. Given $16=2 a e \Rightarrow 16=2 a(\sqrt{2})$
$$
\Rightarrow a=\frac{8}{\sqrt{2}} \Rightarrow a^2=32
$$
Since $b^2=a^2\left(e^2-1\right)=a^2 e^2-a^2=(8)^2-32=32$
$\therefore$ Equation of Hyperbola is
$$
\begin{aligned}
&\frac{x^2}{32}-\frac{y^2}{32}=1 \\
&\Rightarrow x^2-y^2=32
\end{aligned}
$$
$$
\Rightarrow a=\frac{8}{\sqrt{2}} \Rightarrow a^2=32
$$
Since $b^2=a^2\left(e^2-1\right)=a^2 e^2-a^2=(8)^2-32=32$
$\therefore$ Equation of Hyperbola is
$$
\begin{aligned}
&\frac{x^2}{32}-\frac{y^2}{32}=1 \\
&\Rightarrow x^2-y^2=32
\end{aligned}
$$
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