Question

If the distance between the foci and the distance between directrices of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are in the ratio $3:2$, then its eccentricity is ________

Answer 1

Equation of the hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Given that,

$\frac{\text{Distance between foci}}{\text{Distance between directrices}}=\frac{3}{2}\Rightarrow \frac{2ae}{\frac{2a}{e}}=\frac{3}{2}\Rightarrow e^2=\frac{3}{2}\Rightarrow e^2=\frac{3}{2}\Rightarrow e=\sqrt{\frac{3}{2}}$