an ellipse has the following points
(-2,3) left
(2,5) top
(6,3) right
(2,1) bottom
write the equation in general and standard form of the ellipse
The length of the axis are as follows:
Between $\displaystyle \left(-2,3\right),\left(6,3\right)$: length is 6-(-2)=8
Between $\displaystyle \left(2,5\right),\left(2,1\right)$: length is 5-1=4
So it follows that the major axis is parallel to the x axis and $\displaystyle a=4$, and the minor axis is parallel to the y axis and $\displaystyle b=2$.
The ellipse will take the form $\displaystyle \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$.
I leave it for you to verify that the center of the ellipse is (2,3).
Therefore, the equation of the ellipse is $\displaystyle \frac{(x-2)^2}{16}+\frac{(y-3)^2}{4}=1$
Does this make sense?