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(A) Let the hyperbola be $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Given the ratio of the length of the conjugate axis $(2b)$ to the length of the tran

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$13:4$

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(A) Let the hyperbola be $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Given the ratio of the length of the conjugate axis $(2b)$ to the length of the transverse axis $(2a)$ is $3:2$,we have $\frac{2b}{2a} = \frac{3}{2}$,which implies $\frac{b}{a} = \frac{3}{2}$. The distance between the foci is $2ae$. The distance between the two directrices is $\frac{2a}{e}$. The ratio of the distance between the foci to the distance between the directrices is $\frac{2ae}{2a/e} = e^2$. For a hyperbola,$e^2 = 1 + \frac{b^2}{a^2}$. Substituting $\frac{b}{a} = \frac{3}{2}$,we get $e^2 = 1 + (\frac{3}{2})^2 = 1 + \frac{9}{4} = \frac{13}{4}$. Thus,the required ratio is $13:4$.

If for a hyperbola the ratio of the length of the conjugate axis to the length of the transverse axis is $3:2$,then the ratio of the distance between the foci to the distance between the two directrices is

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