Question 451902
You say that the ellipse shares a vertex (the point (0, 100)) and a focus with the parabola {{{x^2 + y = 100}}}, or {{{y = -x^2 + 100}}}, or {{{4*(1/4)*(y - 100) = -(x-0)^2}}}.  Hence for the parabola, the distance from the vertex to the focus is 1/4. Also, one focus of the ellipse must be (0, 399/4).  It was given that the other focus of the ellipse is (0,0) (the origin).  The center of the ellipse is then the point (0, 399/8).  So 

{{{c = 399/4 - 399/8 = 399/8}}}, and {{{a = 100 - 399/8 = 401/8}}}.

Then {{{b^2 = a^2 - c^2 = 25}}}

Then the standard equation  is 

{{{x^2/(401/8)^2 + (y - 399/8)^2/25 = 1}}}, or 

{{{(64x^2)/160801 + (y-399/8)^2/25 = 1}}}.