jmc

From the given information, the center of the ellipse is $(-1,4),$ and the semi-major axis is 10. Thus, the equation of the ellipse is of the form $$\frac{(x+1{)}^2}{10^2}+\frac{(y-4{)}^2}{b^2}=1.$$Setting $x=7$ and $y=7,$ we get $$\frac{8^2}{10^2}+\frac{3^2}{b^2}=1.$$Solving, we find $b^2=25,$ so $b=5.$ Therefore, the area of the ellipse is $\pi \cdot 10\cdot 5=\boxed{50\pi }.$