If the distance between the foci and the distance between the two directricies of the hyperbola x2a2−y2b2=1 are in the ratio 3 : 2, then b : a is

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

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$1 : \sqrt{2}$

$2 : 1$

$\sqrt{3} : \sqrt{2}$

$1 : 2$

answer is A.

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Detailed Solution

Given that $\frac{Distance between foci}{Distance between two directrices} = \frac{3}{2}$

$\begin{array}{l} \Rightarrow \frac{2 ae}{2 \cdot a / e} = \frac{3}{2} \\ \Rightarrow e^{2} = \frac{3}{2} \\ \Rightarrow 1 + \frac{b^{2}}{a^{2}} = \frac{3}{2} \\ \Rightarrow \frac{b}{a} = \frac{1}{\sqrt{2}} \end{array}$