Question

Find the standard form of the equation of the ellipse with the given characteristics.

Answer 1

So hmm if you notice the picture below

between 2 and 16, there are 14 units, so, the center is half-way in between, thus, is at y = 7, or -2, 7

because, the major axis is over the y-axis, then the "a" component, goes under the fraction with the "y" in the numerator

$\bf \textit{ellipse, vertical major axis}\\\\ \cfrac{(x-{{ h}})^2}{{{ b}}^2}+\cfrac{(y-{{ k}})^2}{{{ a}}^2}=1 \qquad center\ ({{ h}},{{ k}})\qquad vertices\ ({{ h}}, {{ k}}\pm a)\\\\ -----------------------------\\\\ \begin{cases} b=4\\ a=7\\ h=-2\\ k=7 \end{cases}\implies \cfrac{(x+2)^2}{4^2}+\cfrac{(y-7)^2}{7^2}\implies \cfrac{(x+2)^2}{16}+\cfrac{(y-7)^2}{49}$