Answer
$\dfrac{x^2}{9}+\dfrac{y^2}{5}=1$
See graph
Work Step by Step
We are given the ellipse:
Foci: $(-2,0),(2,0)$
Length of the major axis: $6$
Because the $y$-coordinates of the foci are the same, the ellipse has the equation:
$\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$
Determine $a$ using the length of the major axis:
$2a=6$
$a=3$
Determine $h,k,c$ from the foci:
$(h-c,k)=(-2,0)\Rightarrow h-c=-2,k=0$
$(h+c,k)=(2,0)\Rightarrow h+c=2$
$\begin{cases}
h-c=-2\\
h+c=2
\end{cases}$
$h-c+h+c=-2+2$
$2h=0$
$h=0$
$h+c=2$
$0+c=2$
$c=2$
Determine $b$:
$a^2=b^2+c^2$
$b^2=a^2-c^2$
$b^2=3^2-2^2$
$b^2=5$
$b=\sqrt{5}$
The equation of the ellipse is:
$\dfrac{x^2}{3^2}+\dfrac{y^2}{(\sqrt 5)^2}=1$
$\dfrac{x^2}{9}+\dfrac{y^2}{5}=1$
The center is:
$(h,k)=(0,0)$
Graph the ellipse:
