Answer
If the foci of the ellipse are located at $\left( -8,6 \right)$ and $\left( 10,12 \right)$; now the coordinate of the center of the ellipse is $\left( 1,9 \right)$.
Work Step by Step
We know that an ellipse can get elongated both horizontally and vertically. And the ellipse is intersected by a line through its foci at two points called vertices. These two vertices are joined by a line segment called the major axis. Now, the midpoint of this major axis is termed as the center of the ellipse. Similarly, the line segment which joins the ellipse vertically and is perpendicular to the major axis at the center is termed in minor axis of the ellipse.
Thus, in the rectangular coordinate system, the formula for getting the midpoint of a line segment with coordinates $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is $M\equiv \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{1}}+{{y}_{2}}}{2}$.
$\begin{align}
& M\equiv \frac{-8+10}{2},\frac{6+12}{2} \\
& \equiv \left( \frac{2}{2},\frac{18}{2} \right) \\
& \equiv \left( 1,9 \right)
\end{align}$
