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  4. If the distance between the foci the distance between the directrices of the hyperbola (x2/a2)-(y2/b2)=1 are in the ratio 3: 2 then a:b is

Q. 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

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Solution:

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

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