Question

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $e$ then the eccentricity of the hyperbola $\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$ is :

• A. $e$
• B. $\frac{e}{\sqrt{e^2-1}}$
• C. $e\sqrt{e^2-1}$
• D. $e^2-e$

Answer 1

Answer: (B)

$\frac{e}{\sqrt{e^2-1}}$

For hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
eccentricity $\Rightarrow e=\sqrt{1+\frac{b^2}{a^2}}\Rightarrow \frac{b^2}{a^2}=e^2-1$
For hyperbola $\frac{x^2}{b^2}-\frac{y^2}{a^2}=1$
Required eccentricity $e^{\prime }=\sqrt{1+\frac{a^2}{b^2}}=\sqrt{1+\frac{1}{e^2-1}}=\frac{e}{\sqrt{e^2-1}}$