If the ratio of the distance between the foci | vedclass

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Question

(A) The distance between the foci is $2ae$. The distance between the directrices is $\frac{2a}{e}$. Given the ratio,we have: $\frac{2ae}{2a/e} = \frac

Vedclass · Accepted Answer

$\sqrt{2} : 1$

Vedclass · Answer

(A) The distance between the foci is $2ae$. The distance between the directrices is $\frac{2a}{e}$. Given the ratio,we have: $\frac{2ae}{2a/e} = \frac{3}{2}$ $\Rightarrow e^2 = \frac{3}{2}$. For a hyperbola,the relationship between $a, b,$ and $e$ is $b^2 = a^2(e^2 - 1)$. Therefore,$\frac{b^2}{a^2} = e^2 - 1$. Substituting $e^2 = \frac{3}{2}$: $\frac{b^2}{a^2} = \frac{3}{2} - 1 = \frac{1}{2}$. Thus,$\frac{b}{a} = \frac{1}{\sqrt{2}}$,which implies $a : b = \sqrt{2} : 1$.

If the ratio of the distance between the foci to the distance between the directrices of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $3 : 2$,then $a : b = \dots$

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