Find the coordinates of the foci and the vertices, the eccentricity,the length of the latus rectum of the hyperbolas : $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1.$
Find the coordinates of the foci and the vertices, the eccentricity,the length of the latus rectum of the hyperbolas : $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1.$
Comparing the equation $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$ with the standard equation $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
Here, $a=3,\,\, b=4$ and $c=\sqrt{a^{2}+b^{2}}=\sqrt{9+16}=5$
Therefore, the coordinates of the foci are $(±5,\,0)$ and that of vertices are $(\pm 3,\,0) .$ Also,
The eccentricity $e=\frac{c}{a}=\frac{5}{3} .$
The latus rectum $=\frac{2 b^{2}}{a}=\frac{32}{3}$
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