If the eccentricity of a hyperbola is $\sqrt{3}$; then the eccentricity of its conjugate hyperbola is :
If the eccentricity of a hyperbola is $\sqrt{3}$; then the eccentricity of its conjugate hyperbola is :
$\sqrt{2}$
$\sqrt{3}$
$\sqrt{\frac{3}{2}}$
$2 \sqrt{3}$
Solution
Let $e$ and $e^{\prime}$ are the eccentricities of a hyperbola and its conjugate hyperbola.
We know that
$\frac{1}{e^2}+\frac{1}{\left(e^{\prime}\right)^2}=1$
$\Rightarrow \quad \frac{1}{3}+\frac{1}{\left(e^{\prime}\right)^2}=1$
$\Rightarrow \quad \frac{1}{\left(e^{\prime}\right)^2}=\frac{2}{3} \Rightarrow e^{\prime}=\sqrt{\frac{3}{2}}$