Question

Find the equation of the hyperbola satisfying the given conditions: Foci $(\pm 5,0)$ the
transverse axis is of length $8$

Answer 1

It is given that, foci $(\pm 5,0),$ the transverse axis is of length $8$
Here the foci are on the $x$-axis.
Therefore, the equation of the hyperbola is of the form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
Since the foci are $(\pm 5,0)\Rightarrow ae=c=5$
Since the length of the transverse axis is $8$, $\Rightarrow 2a=8\Rightarrow a=4$
We know that $a^2+b^2=c^2$
$4^2+b^2=5^2$
$\Rightarrow b^2=25-16=9$
Thus the equation of the hyperbola is $\frac{x^2}{16}-\frac{y^2}{9}=1$