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

Answer 1

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

Distance between foci of a hyperbola is given by $2\sqrt{a^2+b^2}$ and distance between the directrices is given by $\frac{2a^2}{\sqrt{a^2+b^2}}$

$\Rightarrow \frac{2(a^2+b^2)}{2a^2}=\frac{3}{2}$

$\Rightarrow 2a^2+2b^2=3a^2$

$\Rightarrow a^2=2b^2$

$\therefore a:b=\sqrt{2}:1$