Question

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

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

Answer 1

Answer: (A)

$\sqrt{2}:1$

Equation of hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
Distance between foci $=2ae$
and distance between directrices $=\frac{2a}{e}$
According to the question, we have
$\frac{2ae}{2a/e}=\frac{3}{2}$
$\Rightarrow e^2=\frac{3}{2}$
We know that
$b^2=a^2(e^2-1)$
$\Rightarrow \frac{b^2}{a^2}=\frac{3}{2}-1=\frac{1}{2}$
$\Rightarrow \frac{b}{a}=\frac{1}{\sqrt{2}}$
Therefore, required ratio $a:b$ is $\sqrt{2}:1$.