Question

Find the equation of the ellipse whose foci are $(0,\pm 5)$ and the length of whose major axis is 20.

Answer 1

Since the foci of the ellipse lie on the y-axis, it is a vertical ellipse.

Let the required equation be $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1,$ where $a^2>b^2.$

Let $c^2=(a^2-b^2).$

Its foci are $(0,\pm c)$ and therefore, $c=5.$

Also, a = length of the semi-major axis = $(\frac{1}{2}\times 20)=10.$

Now, $c^2=(a^2-b^2)\iff b^2=(a^2-c^2)=(100-25)=75.$

Thus, $a^2=(10{)}^2=100$ and $b^2=75.$

Hence, the required equation is $\frac{x^2}{75}+\frac{y^2}{100}=1.$