# Find the equation for the ellipse that satisfies the given conditions: Length of minor axis 16, foci (0, ±6)

Question:

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

Solution:

Length of minor axis = 16; foci = (0, ±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  b = 8 and c = 6.

It is known that $a^{2}=b^{2}+c^{2}$.

$\therefore a^{2}=8^{2}+6^{2}=64+36=100$

$\Rightarrow a=\sqrt{100}=10$

Thus, the equation of the ellipse is $\frac{x^{2}}{8^{2}}+\frac{y^{2}}{10^{2}}=1$ or $\frac{x^{2}}{64}+\frac{y^{2}}{100}=1$.