Question

Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas. $5y^2-9x^2=36$

Answer 1

Given Hyperbola: $5y^2-9x^2=36$

$\Rightarrow \frac{y^2}{\frac{36}{5}}-\frac{x^2}{4}=1$

where it is a conjugate hyperbola,

$a=2,b=\frac{6}{\sqrt{5}}$

Vertices: $(0,\pm b):(0,\pm \frac{6}{\sqrt{5}})$

Eccentricity: e=$\sqrt{1+\frac{4\left(5\right)}{36\left(9\right)}}=\frac{\sqrt{1}4}{3}$

Foci: $(0,\pm be):(0,\pm \frac{\sqrt{1}4}{3}\cdot \frac{6}{\sqrt{5}})$

$:(0,\pm 2\cdot \sqrt{\frac{14}{5}})$

Length of LR$=\frac{2a^2}{b}=\frac{2\cdot 4\sqrt{5}}{6}=\frac{4}{3}\sqrt{5}$ unit length