Question

Find the equation of the ellipse whose foci are $(0,\pm 4)$ and length of the minor axis is $18$

Answer 1

Since the foci of the ellipse are $F_1(0,-4)$ and $F_2(0,4)$ which lie on the $y-$axis and the mid-point of the line segment $F_1{F}_2$ is $(0,0)$

$\therefore$ origin is the centre of the ellipse and the major axis lies along $y-$axis.

Hence the equation of the ellipse can be taken as

$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ .........$\left(1\right)$

Here $c=4$

Also, the length of the minor axis is $18$

$\Rightarrow 2b=18$

$\Rightarrow b=\frac{18}{2}=9$

We know that $b^2=a^2-c^2$

$\Rightarrow 9^2=a^2-4^2$

$\Rightarrow a^2=81+16=97$

From $\left(1\right)$ , the equation of the ellipse is $\frac{x^2}{81}+\frac{y^2}{97}=1$