Given:
The centre of the ellipse = (-2,4),
A minor axis length a= 12 ,
And distence from the vertex(5,4) to centre ,
[tex]\begin{gathered} b=\sqrt[]{(5-(-2))^2+(4-4)^2} \\ =\sqrt[]{(5+2)^2+0} \\ =\sqrt[]{7^2} \\ =7 \end{gathered}[/tex]The equation of ellipse is ,
[tex]\begin{gathered} \frac{(x-(-2))^2}{7^2}+\frac{(y-4)^2}{12^2}=1 \\ \frac{(x+2)^2}{49}+\frac{(y-4)^2}{144}=1 \\ \end{gathered}[/tex]