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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
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The correct answer is:
$\sqrt{2}: 1$
Equation of hyperbola is
$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
Distance between foci $=2 a e$ and distance between directrices $=\frac{2 a}{e}$ According to question, we have
$\frac{2 a e}{2 a / e}=\frac{3}{2}$
$\begin{array}{ll}\Rightarrow & e^{2}=\frac{3}{2} \\ \because & b^{2}=a^{2}\left(e^{2}-1\right) \\ \Rightarrow & \frac{b^{2}}{a^{2}}=\frac{3}{2}-1=\frac{1}{2} \\ \Rightarrow & \frac{b}{a}=\frac{1}{\sqrt{2}} \\ \Rightarrow & a: b=\sqrt{2}: 1\end{array}$
$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
Distance between foci $=2 a e$ and distance between directrices $=\frac{2 a}{e}$ According to question, we have
$\frac{2 a e}{2 a / e}=\frac{3}{2}$
$\begin{array}{ll}\Rightarrow & e^{2}=\frac{3}{2} \\ \because & b^{2}=a^{2}\left(e^{2}-1\right) \\ \Rightarrow & \frac{b^{2}}{a^{2}}=\frac{3}{2}-1=\frac{1}{2} \\ \Rightarrow & \frac{b}{a}=\frac{1}{\sqrt{2}} \\ \Rightarrow & a: b=\sqrt{2}: 1\end{array}$
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