Class 11

Math

3D Geometry

Conic Sections

Find the coordinates of the foci, the vertices the eccentricity and the length of latus rectum of the hyperbola $9y_{2}−4x_{2}=36$

It can be written as

$9y_{2}−4x_{2}=36$

$4y_{2} −9x_{2} =1$

$2_{2}y_{2} −3_{2}x_{2} =1...(1)$

On comparing equation (1) with the standard equation of hyperbola

i.e., $a_{2}y_{2} −b_{2}x_{2} =1$

we obtain $a=2$ and $b=3$

We know that $a_{2}+b_{2}=c_{2},$ where $c=ae$

$∴c_{2}=4+9=13$

$⇒c=13 $

Therefore, the coordinates of the foci are $(0,± 13 )$

The coordinates of the vertices are $(0,±2)$

Eccentricity $e$ $=ac =213 $

Length of latus rectum $=$ $a2b_{2} =22×9 =9$