The vertices of the ellipse are given as
[tex]\begin{gathered} V_2=(4,-3) \\ V_1=(4,7) \end{gathered}[/tex]The focus of the ellipse is given as
[tex](4,4)[/tex]The equation of an ellipse is given as
[tex]\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1[/tex]Where the coordinate of the center is
[tex](h,k)[/tex]To calculate the coordinate of the center, we will use the formula below
[tex]\begin{gathered} \frac{(x_1+x_2)}{2},\frac{(y_1+y_2)}{2} \\ =\frac{(4+4)}{2},\frac{(-3+7)}{2} \\ =\frac{8}{2},\frac{4}{2} \\ (4,2) \\ (h,k)=(4,2) \end{gathered}[/tex]The formula to calculate the value of a is given below
[tex]\begin{gathered} V_1=(h,k+a) \\ V_2=(h,k-a) \end{gathered}[/tex]By comparing coefficients, we will have
[tex]\begin{gathered} k+a=7\ldots\text{.}(1) \\ k-a=-3\ldots\text{.}(2) \end{gathered}[/tex]By adding equations (1) and (2) and solving simultaneously, we will have
[tex]\begin{gathered} 2k=4 \\ \text{divide both sides by 2,} \\ \frac{2k}{2}=\frac{4}{2} \\ k=2 \end{gathered}[/tex]By substituting hk=2 in equation 1, we will have
[tex]\begin{gathered} k+a=7 \\ 2+a=7 \\ a=7-2 \\ a=5 \end{gathered}[/tex]The coordinate of the focus,is calculated using the formula below
[tex](h,k+c)[/tex]By substituting the values, we will have
[tex]\begin{gathered} k+c=4 \\ 2+c=4 \\ c=4-2 \\ c=2 \end{gathered}[/tex]The value of will be calculated using the formula below
[tex]\begin{gathered} c^2=a^2-b^2 \\ b^2=a^2-c^2 \end{gathered}[/tex]By substituting the values, we will have
[tex]\begin{gathered} b^2=a^2-c^2 \\ b^2=(5)^2-(2^2 \\ b^2=25-4 \\ b^2=21 \end{gathered}[/tex]By substituting the values of a,b,h and k in the equation of an ellipse, we will have
[tex]\begin{gathered} \frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1 \\ \frac{(x-4)^2}{21^{}}+\frac{(y-2)^2}{25^{}}=1 \\ \end{gathered}[/tex]Hence,
The equation of the ellipse will be
[tex]\frac{(x-4)^2}{21^{}}+\frac{(y-2)^2}{25^{}}=1[/tex]