Hocus, Pocus, LOCUS!

If the locus of the focus of the variable parabola, which always touches the curve $C_2$ : $x^2=-y$ at $(0,0)$ , is $C_1$ : $x^2 + y^2 = 2ay^4$ .

If $C_1$ and $C_2$ have same length of Latus Rectum and they have no other intersection point, find $a$ .

Credits: My Mathematics Teacher.