Question

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

(a) x² – y² = 32

(b) $\frac{x^2}{4}-\frac{y^2}{9}=1$

(c) 2x² – 3y² = 7

(d) none of these

Answer 1

$$\begin{gathered} \mathrm{For}\mathrm{an}\mathrm{hyperbola}, \\ \mathrm{eccentricity}\mathrm{is}\sqrt{2}\left(\mathrm{given}\right) \\ \mathrm{Distance}\mathrm{between}\mathrm{foci}\mathrm{is}16 \\ \mathrm{i}.\mathrm{e}.2ae=16 \\ \mathrm{i}.\mathrm{e}.2\times a\times \sqrt{2}=16 \\ \mathrm{i}.\mathrm{e}.a=\frac{8}{\sqrt{2}}\times \frac{\sqrt{2}}{2}=4\sqrt{2}\mathrm{i}.\mathrm{e}.a^2=32 \\ \therefore b^2=32\left(2-1\right) \\ {b}^2=32 \\ \therefore \mathrm{equation}\mathrm{of}\mathrm{hyperbola}\mathrm{is} \\ \frac{x^2}{32}-\frac{y^2}{32}=1 \\ \mathrm{i}.\mathrm{e}.x^2-y^2=32 \end{gathered}$$
Hence, the correct answer option is A.