Question

Find the equation for the ellipse that given that satisfies the given conditions: Length of minor axis $16$, foci $(0,\pm 6).$

Answer 1

It is given that, length of minor axis $=16$; foci = $(0,\pm 6)$
Since the foci are on the $y$-axis the major axis is along the $y$-axis
Therefore, the equation of the ellipse will be of the form $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ where $a$ is the semi-major axis
Accordingly $2b=16,$ $\Rightarrow b=8$ and $ae=6$
We know that, $a^2=b^2+a^2{e}^2$
$\therefore a^2=8^2+6^2=64+36=100$
$\Rightarrow a=\sqrt{100}=10$
Thus, the equation of the ellipse is $\frac{x^2}{8^2}+\frac{y^2}{{10}^2}=1$or $\frac{x^2}{64}+\frac{y^2}{100}=1$