Question

Find the equation for the ellipse that satisfies the given conditions:

Vertices $(\pm 6,0)$, foci $(\pm 4,0).$

Answer 1

It is given that, vertices $(\pm 6,0)$, foci $(\pm 4,0)$
Clearly, the vertices are on the $x$-axis.
Therefore, the equation of the ellipse will be of the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where $a$ is the semi-major axis.
Also, $a=6,ae=4\Rightarrow e=\frac{2}{3}$
We know that $b^2=a^2(1-e^2)=a^2-a^2{e}^2=36-16=20$
Thus, the equation of the ellipse is $\frac{x^2}{36}+\frac{y^2}{20}=1$