It seems the student is trying to write the equation of a standard parabola.
A parabola is defined as the locus of points equidistant from a focus point #(p,q)# and a directrix line. For standard parabolas, the directrix is of the form #y=k#. We can tell it's a standard parabola here because the vertex and focus have the same #x# coordinate.
The squared distance from the point #(x,y)# to the directrix #y=k# is of course #(y-k)^2#. The squared distance from the point #(x,y)# to the focus is #(x-p)^2 + (y-q)^2#. To find our parabola equation, we equate those:
#(y-k)^2= (x-p)^2 + (y-q)^2#
# y^2 - 2k y + k^2 = x^2 - 2px + p^2 + y^2 - 2qy + q^2 #
#2(q-k) y = x^2 - 2px + q^2 - k^2#
# y = \frac{1]{ 2(q-k) ) (x^2 - 2px + (q^2-k^2)) #
That's the general solution, let's plug in our numbers. #(p,q)#=(0,4). Our directrix must be equidistant from our focus, so must pass through (0,-4), so must be #y=-4# so #k=-4#.
# y = frac{1}{16} x^2 #
All the other terms vanish.
Check: Let's check at #x=4#. We get #y=1#. The distance of #(4,1)# to the focus #(0,4)# is #sqrt{4^2 + 3^2 } = 5#. The distance of (4,1) to #y=-4# is also #5. \quad \sqrt{}#
