Question

Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in  the  conjugate axis is 5 and the distance between foci = 13 .

Answer 1

The distance between the foci is \[2ae\] .

\[\therefore 2ae = 13\]

\[ \Rightarrow ae = \frac{13}{2}\]

Length of the conjugate axis,

\[2b = 5\]

\[\Rightarrow b = \frac{5}{2}\]

Also, \[b^2 = a^2 ( e^2 - 1)\]

\[ \Rightarrow \left( \frac{5}{2} \right)^2 = \left( \frac{13}{2} \right)^2 - a^2 \]

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

\[ \Rightarrow a^2 = \frac{144}{4} = 36\]

\[ \Rightarrow a = 6\]

Therefore, the standard form of the hyperbola is \[\frac{x^2}{36} - \frac{4 y^2}{25} = 1\] . 

\[or 25 x^2 - 144 y^2 = 900\]