Question

Find the equation of the hyperbola ( with foci along $x-axis$), the length of whose latus rectum is $8$ and eccentricity is $\frac{3}{\sqrt{5}}$

Answer 1

Length of latus rectum $=8$

$\begin{array}{c}\Rightarrow \frac{2b^2}{a}=8\Rightarrow \frac{b^2}{a}=4b^2=4a\\ b^2=a^2\left(e^2-1\right)\\ 4a=a^2\left(\frac{a}{5}-1\right)\\ =a^2\left(\frac{4}{5}\right)\\ \Rightarrow a=\frac{a^2}{5}\\ \Rightarrow \frac{a^2}{5}-a=0\\ a\left(\frac{a}{5}-1\right)=0\\ a=0ora=5\therefore b^2=20\\ Equationofhyperbola,\\ \frac{x^2}{25}+\frac{y^2}{20}=1\end{array}$