Question

Find the equation of the ellipse whose foci are $(0,\pm 1)$ and length of the minor axis is $12$

Answer 1

Since the foci of the ellipse are $F_1(0,-1)$ and $F_2(0,1)$ 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 $c=1$

Also, length of the minor axis is $12$

$\Rightarrow 2b=12$

$\Rightarrow b=\frac{12}{2}=6$

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

$\Rightarrow 6^2=a^2-1^2$

$\Rightarrow a^2=36+1=37$

From $\left(1\right)$ , the equation of the ellipse is $\frac{x^2}{36}+\frac{y^2}{37}=1$