Find the coordinates of the foci and the vertices, the eccentricity,the length of the latus rectum of the hyperbolas : $y^{2}-16 x^{2}=16$
Find the coordinates of the foci and the vertices, the eccentricity,the length of the latus rectum of the hyperbolas : $y^{2}-16 x^{2}=16$
Dividing the equation by $16$ on both sides, we have $\frac{y^{2}}{16}-\frac{x^{2}}{1}=1$
Comparing the equation with the standard equation $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1,$ we find that $a=4, b=1$ and $c=\sqrt{a^{2}+b^{2}}=\sqrt{16+1}=\sqrt{17}$
Therefore, the coordinates of the foci are $(0, \,\pm \sqrt{17})$ and that of the vertices are $(0,\,\pm 4) .$ Also,
The eccentricity $e=\frac{c}{a}=\frac{\sqrt{17}}{4} .$
The latus rectum $=\frac{2 b^{2}}{a}=\frac{1}{2}$
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