Question

If $e_1$ is the eccentricity of the ellipse $\frac{x^2}{16}+\frac{y^2}{25}=1$ and $e_2$ is the eccentricity of the hyperbola passing through the foci of the ellipse and $e_1{e}_2=1$, then equation of the hyperbola is

• A. $\frac{x^2}{9}-\frac{y^2}{16}=1$
• B. $\frac{x^2}{16}-\frac{y^2}{9}=-1$
• C. $\frac{x^2}{9}-\frac{y^2}{25}=1$
• D. None of these

Answer 1

The correct option is B

$\frac{x^2}{16}-\frac{y^2}{9}=-1$

The eccentricity of $\frac{x^2}{16}+\frac{y^2}{25}=1$ is
$e_1=\sqrt{1-\frac{16}{25}}=\frac{3}{5}$
$\therefore e_2=\frac{5}{3}$ $[\because e_1{e}_2=1]$
$\Rightarrow$ Foci of ellipse $(0,\pm 3)$
$\Rightarrow$ Equation of hyperbola is $\frac{x^2}{16}-\frac{y^2}{9}=-1$.