Question

If the distance between the foci & 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. $\sqrt{3}: \sqrt{2}$
• B. $\sqrt{2}: 1$
• C. $2: 1$
• D. $1: 2$

Answer 1

Answer: (B)

$\frac{\text { Distance between Foci }}{\text { Distance between Directrix }}=\frac{3}{2}$
$\frac{2 a e}{\frac{2 a}{e}}=\frac{3}{2} e^{2}=3 / 2$
$1+\frac{b^{2}}{a^{2}}=\frac{3}{2}$
$\frac{b^{2}}{a^{2}}=\frac{3}{2}-1$
$\frac{b^{2}}{a^{2}}=\frac{1}{2}\,\, \frac{b}{a}=\frac{1}{\sqrt{2}}$
$\frac{a}{b}=\frac{\sqrt{2}}{1}$
$\therefore a: b=\sqrt{2}: 1$