Question

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

Answer 1

Given Hyperbola: $16x^2-9y^2=576$

Dividing it by 576, we get $\frac{x^2}{36}-\frac{y^2}{64}=1$

where $a=6,b=8$

Vertices A$(\pm a,0):(\pm 6,0)$

Eccentricity, e$=\sqrt{1+\frac{b^2}{a^2}}=\sqrt{1+\frac{64}{34}}=\frac{10}{6}=\frac{5}{3}$

Foci :$(\pm ae,0):(\pm \frac{5}{3}\cdot 6,0):(\pm 10,0)$

Length of LR$=2\frac{b^2}{a}=2\cdot \frac{64}{6}=\frac{64}{3}$ unit length.