Question

Find the equation of a hyperbola in the standard equation whose conjugate axis is $5$ and the distance between the foci is $13$.

Answer 1

Conjugate axis $=5$

$2b=5$

$b=5/2$

distance between foci $=13$

$\begin{array}{c}2ae=13\\ ae=\frac{13}{2}\\ b^2=a^2\left(e^2-1\right)\\ {\left(ae\right)}^2-a^2={\left(\frac{13}{2}\right)}^2-a^2\\ =a^2=\frac{169-b^2}{4}=\frac{169}{4}-\frac{25}{4}=\frac{144}{4}\\ a=\frac{12}{2}=6\\ \therefore equationofhyperbola,\\ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\\ \frac{x^2}{36}-\frac{y^2}{25}=1\end{array}$