Question

Let the eccentricity of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{ b^2} = 1 $ be reciprocal to that of the ellipse $x^2 + 9y^2 = 9$, then the ratio $a^2 : b^2$ equals

• A. 8 : 1
• B. 1 : 8
• C. 9 : 1
• D. 1 : 9

Answer 1

Answer: (A)

Given equation of ellipses is $x^{2}+9 y^{2}=9$
$\Rightarrow \frac{x^{2}}{9}+\frac{y^{2}}{1}=1$
Here, $a=3, b=1$
$c=\sqrt{(3)^{2}-(1)^{2}}=\sqrt{8}$
$\therefore $ Eccentricity of ellipse, $e=\frac{c}{a}$
$\Rightarrow e=\frac{\sqrt{8}}{3}$
$\therefore $ Eccentricity of hyperbola $=\frac{3}{\sqrt{8}}$
$\Rightarrow 1+\frac{b^{2}}{a^{2}}=\frac{9}{8}$
$\Rightarrow \frac{b^{2}}{a^{2}}=\frac{1}{8}$
$\Rightarrow a^{2}: b^{2}=8: 1$