Question

The distance between the foci of a hyperbola is 16 and its eccentricity is $\sqrt{2}$, then equation of the hyperbola is
(a) x² + y² = 32
(b) x² − y² = 16
(c) x² + y² = 16
(d) x² − y² = 32

Answer 1

(d) x² − y² = 32

The distance between the foci is $2ae$.

$$\begin{gathered} \therefore 2ae=16 \\ \Rightarrow ae=8 \end{gathered}$$

$e=\sqrt{2}$
$$\begin{gathered} \therefore a\sqrt{2}=8 \\ \Rightarrow a=4\sqrt{2} \end{gathered}$$


$$\begin{gathered} \text{Also,}{b}^2=a^2(e^2-1) \\ \Rightarrow b^2=32(2-1) \\ \Rightarrow b^2=32 \end{gathered}$$
Standard form of the hyperbola is given below:
$$\begin{gathered} \frac{x^2}{32}-\frac{y^2}{32}=1 \\ {x}^2-y^2=32 \end{gathered}$$