Question

What is the equation of the eclipse with focus (0,7),(0,-7) and vertices (0,8),(0,-8)

Answer 1

check the picture below.

so the ellipse looks more or less like that, notice the "a" component is 8 units, whilst the "c" component, namely the distance from the center to a focus is 7 units, thus


`\bf \textit{ellipse, vertical major axis} \\\\ \cfrac{(x- h)^2}{ b^2}+\cfrac{(y- k)^2}{ a^2}=1 \qquad \begin{cases} center\ ( h, k)\\ vertices\ ( h, k\pm a)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad √( a ^2- b ^2) \end{cases}\\\\ -------------------------------`


`\bf \begin{cases} h=0\\ k=0\\ a=8\\ c=7 \end{cases}\implies \cfrac{(x- 0)^2}{ b^2}+\cfrac{(y- 0)^2}{ 8^2}=1 \\\\\\ c=√(a^2-b^2)\implies c^2=a^2-b^2\implies b^2=a^2-c^2 \\\\\\ b^2-64-49\implies b^2=15\qquad \qquad \cfrac{(x- 0)^2}{ 15}+\cfrac{(y- 0)^2}{ 8^2}=1 \\\\\\ \cfrac{x^2}{15}+\cfrac{y^2}{64}=1`