Question

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $9y^2-4x^2=37$

Answer 1

The given equation is $9y^2 - 4x^2 = 36.$
or $\dfrac{y^2}{4} – \dfrac{x^2}{9} = 1$

or $\dfrac{y^2}{2^2} – \dfrac{x^2}{3^2} = 1.......(1)$
On comparing equation (1) with the standard equation of hyperbola i.e., $\dfrac{y^2}{a^2} – \dfrac{x^2}{b^2} = 1$ we obtain $a = 2$ and $b = 3$. $$
We know that $a^2 + b^2 = c^2 .$
$∴ c^2 = 4 + 9$
$c^2 = 13$
$c = √13.$

Therefore,
The coordinates of the foci are $(0, √13)$ and $(0, –√13).$
The coordinates of the vertices are $(0, 2)$ and $(0, – 2).$
Eccentricity, $e = \dfrac{c}{a} = \dfrac{√13}{2}$
Length of the latus rectum $= \dfrac{2b^2}{a} = \dfrac{(2 × 3^2)}{2}$

$= \dfrac{(2×9)}{2} = \dfrac{18}{2} = 9$