Question

Find the equation of the hyperbola whose Transverse and Conjugate axes are the $x$ and $y$ axes respectively, given that the length of conjugate axis is $5$ and distance between the foci is $13$.

Answer 1

Let equation of hyperbola be $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ ...... (i)
According to question, $2b=5\Rightarrow b=5/2$
$2ae=13\Rightarrow ae=13/2$
Also, $b^2=a^2{e}^2-a^2$
${\left(\frac{5}{2}\right)}^2={\left(\frac{13}{2}\right)}^2-a^2$
$\Rightarrow a^2={\left(\frac{13}{2}\right)}^2-{\left(\frac{5}{2}\right)}^2$
$\Rightarrow a^2=36$
Substituting values of $a^2$ and $b^2$ in equation (i),
$\frac{x^2}{36}-\frac{y^2}{\frac{25}{4}}=1$
$\Rightarrow \frac{x^2}{36}-\frac{4y^2}{25}=1$
$\Rightarrow \frac{25x^2-144y^2}{900}=1\Rightarrow 25x^2-144y^2=900$