I followed my professor's solution method, and calculated C = po + ρGM/R1, R1 being the small radius of the Earth's ellipsoid, then, I made p(x,z) = po, which led me to GM/R1 = w^2x^2/2 + GM/√(x^2 + z^2)
I was hopping that the ellipse equation would emerge naturally (as the circle equation did...