Question

Find the equation of the ellipse whose foci are $(0,\pm 7)$ and length of the minor axis is $30$

Answer 1

Since the foci of the ellipse are $F_1(0,-7)$ and

$F_2(0,7)$ 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 $c=7$

Also, length of the minor axis is $30$

$\Rightarrow 2b=30$

$\Rightarrow b=\frac{30}{2}=15$

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

$\Rightarrow {15}^2=a^2-7^2$

$\Rightarrow a^2=225+49=274$

From $\left(1\right)$ , the equation of the ellipse is $\frac{x^2}{225}+\frac{y^2}{274}=1$