Question

Find the coordinates of the foci, the vertices the eccentricity and the length of latus rectum of the hyperbola $16x^2-9y^2=576$

Answer 1

We have, $16x^2-9y^2=576$
$\Rightarrow \frac{x^2}{36}-\frac{y^2}{64}=1$
$\Rightarrow \frac{x^2}{6^2}-\frac{y^2}{8^2}=1...\left(1\right)$
On comparing equation (1) with the standard equation of hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,
we obtain $a=6,b=8\Rightarrow e=\sqrt{1+\frac{b^2}{a^2}}=\sqrt{1+\frac{64}{36}}=\frac{5}{3}$
Therefore, the coordinates of the foci are $(\pm 10,0)$
The coordinates of the vertices are $(\pm 6,0)$
Eccentricity $e=\frac{5}{3}$
Length of latus rectum $=$ $\frac{2b^2}{a}=\frac{2\times 64}{6}=\frac{64}{3}$