Class 11

Math

3D Geometry

Conic Sections

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $36x_{2}+4y_{2}=144$

It can be written as

$36x_{2}+4y_{2}=144$

$4x_{2} +36y_{2} =1$

$2_{2}x_{2} +6_{2}y_{2} =1$ ...(1)

Here the donominator of $6_{2}y_{2} $ is greater than the denominator of $2_{2}x_{2} $

Therefore, the major axis is along the $y$-axis while the minor axis is along the $x$-axis.

On comparing equation (1) with $b_{2}x_{2} +a_{2}y_{2} =1$, we obtain $b=2$ and $a=6$

$∴ae=c=a_{2}−b_{2} =36−4 =32 =42 $

Therefore, the coordinates of the foci are $(0,±42 )$

The coordinates of the vertices are (0, $±6)$

Length of major axis $=2a=12$

Length of minor axis $=2b=4$

Eccentricity $e=$ $ac =642 =322 $

Length of latus rectum $=$ $a2b_{2} =62×4 =34 $