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If the transverse and conjugate axes of hyperbola are equal, then its eccentricity is
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Verified Answer
The correct answer is:
$\sqrt{2}$
Given that, transverse and conjugate axis of hyperbola are equal.
$C D \rightarrow$ transverse axis
$A B \rightarrow$ conjugate axis
$$
\Rightarrow \quad 2 b=2 a \Rightarrow a=b
$$
Using equation of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
$$
\left\{\because b^2=a^2\left(e^2-1\right)\right\}
$$
Now, $a^2=a^2\left(e^2-1\right) \Rightarrow e^2=2$
$$
\Rightarrow \quad e=\sqrt{2}
$$
$C D \rightarrow$ transverse axis
$A B \rightarrow$ conjugate axis
$$
\Rightarrow \quad 2 b=2 a \Rightarrow a=b
$$
Using equation of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
$$
\left\{\because b^2=a^2\left(e^2-1\right)\right\}
$$
Now, $a^2=a^2\left(e^2-1\right) \Rightarrow e^2=2$
$$
\Rightarrow \quad e=\sqrt{2}
$$
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