Question

Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in following cases :
Conjugate axis is $5$ and distance between foci $=13$

Answer 1

Given that, Conjugate axis $=5$

$\Rightarrow 2b=5$

$\Rightarrow b=\frac{5}{2}$

$\Rightarrow b^2=\frac{25}{4}\dots \left(1\right)$

$\because$ Distance between the foci $=13$

$\Rightarrow 2ae=13$

$\Rightarrow ae=\frac{13}{2}$

$\because b^2=a^2{e}^2-a^2$

$\Rightarrow \frac{25}{4}=\frac{169}{4}-a^2$

$\Rightarrow a^2=\frac{169}{4}-\frac{25}{4}$

$\Rightarrow a^2=\frac{144}{4}\dots \left(2\right)$

Equation of hyperbola when axes as axes of coordinate

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$\Rightarrow \frac{x^2}{\frac{144}{4}}-\frac{y^2}{\frac{25}{4}}=1$

$\Rightarrow \frac{4x^2}{144}-\frac{4y^2}{25}=1$

$\Rightarrow 4\left(\frac{25x^2-144y^2}{25\times 144}\right)=1$

$\Rightarrow 25x^2-144y^2=900$