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

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Given $\; 16x^2-9y^2=576.$ Dividing by $576 \rightarrow \large\frac{16}{576}$$a^2-\large\frac{9}{576}$$y^2=1 \rightarrow \large\frac{x^2}{36} $$-\large\frac{y^2}{64}$$=1$
On comparing the given equation with the standard equation of a hyperbola, we get:
$\quad a = 6, b = 8$
Since $a^2+b^2 = c^2 \rightarrow c = \sqrt{36+64} = \sqrt {100} = 10$
1. Coordinates of Foci $= (\pm 10, 0)$
2. Coordinates of Vertices $= (\pm 6, 0)$
3. Eccentricity $e = \large\frac{c}{a} $$=\large\frac{10}{6}$$=\large\frac{5}{3}$
4. Latus Rectum $=\large\frac{2b^2}{a}$$= \large\frac{2\times 8^2}{6}$$= \large \frac {64}{3}$