Question

If the eccentricity of a hyperbola is $\frac{5}{4}$, then find the eccentricity of its conjugate.

Answer 1

Fact: Sum of square of reciprocal of eccentricities of hyperbola and its conjugate is $1$

Let eccentricity of the conjugate is $e$, then

$\frac{1}{e^2}+\frac{16}{25}=1$

$\Rightarrow \frac{1}{e^2}=1-\frac{16}{25}=\frac{9}{25}$

$\Rightarrow e^2=\frac{25}{9}$

$\therefore e=\frac{5}{3}$