Question

Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola $49y^2-16x^2=784$

Answer 1

Given: Equation of hyperbola is $49y^2-16x^2=784$
i.e, $\frac{y^2}{16}-\frac{x^2}{49}=1$

On comparing the given equation with $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ we get

$a^2=16\Rightarrow a=4$ and $b^2=49\Rightarrow b=7$

$\Rightarrow c=\sqrt{(a^2+b^2)}$

$=\sqrt{(16+49)}=\sqrt{65}$

Then,
The coordinates of the foci $=(0,\pm c)=(0,\pm \sqrt{65})$

$a=4,b=7,c=\sqrt{65}$

The coordinates of the vertices $=(0,\pm a)=(0,\pm 4)$

Eccentricity, $e=\frac{c}{a}=\frac{\sqrt{65}}{4}$

Length of latus rectum $=\frac{2b^2}{a}$
$=\frac{2\times 49}{4}$
$=\frac{49}{2}$