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

Question

Dr. Abubacker · Accepted Answer

(a)
Equation of hyperbola is $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\ \$
Distance between foci of hyperbola$= 2ae\$
and its distance between directrices $= \frac{2a}{e}$
According to the question,
$\frac{2ae}{2a/e} = \frac{3}{2}$
$\Rightarrow \ e^{2} = \frac{3}{2}$
Using , $b^{2}\  = a^{2}\left( e^{2}\  - 1 \right)\ \  \Rightarrow \ \ \frac{b^{2}}{a^{2}} = \frac{3}{2}\  - 1$
$\Rightarrow \ \ \frac{a}{b} = \frac{\sqrt{2}}{1}$

Q. 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

Explanation

Reviewed by Dr. Abubacker

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