Question

Find the coordinates of the foci, the vertices, the accentricity and the length of the latus rectum of the hyperbola,

$\frac{y^2}{9}-\frac{x^2}{27}=1$

Answer 1

The equation of given hyperbola is,
$\frac{y^2}{9}-\frac{x^2}{27}=1$ Which is of the form $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$
The foci and vertices of the hyperbola lie on x - axis.
$\therefore a^2=9\Rightarrow a=3\text{and}{b}^2=27\Rightarrow b=3\sqrt{3}\text{Now}{c}^2=a^2+b^2=9+27=36\Rightarrow c=6\therefore \text{Coordinates of foci are}(0,\pm c)i.e.(0,\pm 6)\text{Coordinates of vertices are}(0,\pm a)i.e.(0,\pm 3)\text{Eccentricity}\left(e\right)=\frac{c}{a}=\frac{6}{3}=2\text{Length of latus rectum}=\frac{2b^2}{a}=\frac{2\times 27}{3}=18$