Question

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

• A. $1:\sqrt{2}$
• B. $\sqrt{3}:\sqrt{2}$
• C. $1:2$
• D. $2:1$

Answer 1

The correct option is A $1:\sqrt{2}$

Distance between foci $=2ae$
Distance between directrices $=\frac{2a}{e}$
$\frac{2a{e}^2}{2a}=\frac{3}{2}$
$\Rightarrow e^2=\frac{3}{2}$
$\Rightarrow 1+\frac{b^2}{a^2}=\frac{3}{2}$
$\Rightarrow \frac{b^2}{a^2}=\frac{1}{2}$
$\Rightarrow \frac{b}{a}=\frac{1}{\sqrt{2}}$