Whoops, I realized that the problem is using a different form of the vertical hyperbola instead of [tex]\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1[/tex], so in this case switch a and b. In the other problem, where b=10 is correct.
Since (h,k) in the parabolic model is the vertex whereas the (h,k) in the semi-elliptical model is the center, I believe you should also compare the focus of the parabola and the focii of the ellipse.
The focii for an ellipse is the point that lies on the major axis(the longer side/axis) of the ellipse. There are two focii in this ellipse. (h+c,k) and (h-c,k). In an ellipse(for both vertial and horizontal ellipses), [tex]b^2=a^2-c^2[/tex], where a is always the large axis and b is the smaller axis.
In a parabola, the there is only one focus. Since this parabola opens down, then the focus is at (h,-c+k).