Question

Find the equation of the ellipse with the following properties.

The ellipse with foci at (4,0) and (-4,0); y-intercepts (0,3) and (0, -3).

Answer 1

check the picture below, so the ellipse looks more or less like so.

since the major axis this time is over the x-axis, yeap, you guessed it, "a" will be under the "x" fraction.

notice the graph, we know b = 3, and c = 4.


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


`\bf \begin{cases} h=0\\ k=0\\ b=3\\ c=4 \end{cases}\implies \cfrac{(x- 0)^2}{ a^2}+\cfrac{(y- 0)^2}{ 3^2}=1 \\\\\\ c=√( a ^2- b ^2)\implies c^2=a^2-b^2\implies 4^2=a^2-3^2 \\\\\\ 4^2+3^2=a^2\implies √(4^2+3^2)=a\implies \boxed{5=a} \\\\\\ \cfrac{(x- 0)^2}{ 5^2}+\cfrac{(y- 0)^2}{ 3^2}=1\implies \cfrac{x^2}{25}+\cfrac{y^2}{9}=1`