Question

Find an equation of the ellipse having a major axis of length 8 and foci at (1, 2) and (1, -2).

Answer 1

Check the picture below.

from the foci points we can see more or less that the traverse axis of the ellipse is a vertical one, since one is above and the other below, and in between equidistantly is the center of the ellipse at (h , k), which we can pretty much get to be (1 , 0).

we know the major axis is 8, namely the "a" component is half that or 4, as you can see in the picture, the ellipse looks more or less like so, since the ellipse has a vertical major axis, that means that "a" goes below the fraction with the "y" in it.

we also know that the distance "c" from a focus point to the vertex is 2, so

`\textit{ellipse, vertical major axis} \\\\ \cfrac{(x- h)^2}{ b^2}+\cfrac{(y- k)^2}{ a^2}=1 \quad \begin{cases} center\ ( h, k)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad √( a ^2- b ^2)\\[-0.5em] \hrulefill\\ h=1\\ k=0\\ a=4\\ c=2 \end{cases} ~\hfill \cfrac{(x-1)^2}{b^2}+\cfrac{(y-0)^2}{4^2}=1 \\\\\\ \stackrel{c}{2}=√(4^2-b^2)\implies 4=16-b^2\implies b^2+4=16 \implies b^2=12 \\\\[-0.35em] ~\dotfill\\\\ ~\hfill \cfrac{(x-1)^2}{12}+\cfrac{y^2}{16}=1~\hfill`