Answer
The graph of $\frac{{{\left( x+1 \right)}^{2}}}{25}+\frac{{{\left( y-4 \right)}^{2}}}{9}=1$ has its center at $\left( -1,4 \right)$.
Work Step by Step
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. And the midpoint of this major axis is termed in center of the ellipse. Similarly, the line segment joining the ellipse vertically is termed as the minor axis of the ellipse.
The standard form of the equation of an ellipse centered at $\left( h,k \right)$ is $\frac{{{\left( x-h \right)}^{2}}}{{{a}^{2}}}+\frac{{{\left( y-k \right)}^{2}}}{{{b}^{2}}}=1$ or $\frac{{{\left( x-h \right)}^{2}}}{{{b}^{2}}}+\frac{{{\left( y-k \right)}^{2}}}{{{a}^{2}}}=1$.
In the standard form of the equation centered at $\left( h,k \right)$, h is the number subtracted from x and k is the number subtracted from y.
$\frac{{{\left( x-\left( -1 \right) \right)}^{2}}}{25}+\frac{{{\left( y-4 \right)}^{2}}}{9}=1$
Thus, $h=-1$ and $k=4$. Hence, the center of the ellipse, $\left( h,k \right)$, is $\left( -1,4 \right)$.
