We need to find the equation of the ellipse given the foci and major axis.
The equation is given by:
[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex]We have the foci points at (2,4) and (2,-6). The distance is 10 units, then the center is equal to half value c=5.
If the length of the major axis is 12.
Major axis length = 2a
Then
12 = 2a
a=12/2
a=6
To find b, we have the next equation:
[tex]\begin{gathered} b^2=a^2-c^2 \\ Replacing \\ b^2=6^2-5^2 \\ b^2=36-25 \\ b^2=11 \\ Solve\text{ for b} \\ b=\sqrt[]{11} \end{gathered}[/tex]Now, the center c is given by the point (2,-1). This point is in the middle of both foci points
Finally, we can replace on the ellipse equation:
[tex]\begin{gathered} \frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1 \\ Replacing,\text{ the result is} \\ \frac{(x-2)^2}{6^2}+\frac{(y-(-1))^2}{(\sqrt[]{11})^2}=1 \end{gathered}[/tex]