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

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Solution

Conjugate axis $= 5$
$2 b = 5$
$b = 5 / 2$
distance between foci $= 13$
$\begin{matrix} 2 a e = 13 a e = \frac{13}{2} b^{2} = a^{2} (e^{2} - 1) {(a e)}^{2} - a^{2} = {(\frac{13}{2})}^{2} - a^{2} = a^{2} = \frac{169 - b^{2}}{4} = \frac{169}{4} - \frac{25}{4} = \frac{144}{4} a = \frac{12}{2} = 6 ∴ e q u a t i o n o f h y p e r b o l a, \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \frac{x^{2}}{36} - \frac{y^{2}}{25} = 1 \end{matrix}$

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

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