Focus of conic formed by intersection of a plane with any enveloping cone of a given sphere

Question

Show that the plane $z= a$ meets any enveloping cone of $x^2+y^2+z^2= a^2$ in a conic which has a focus at the point $(0,0,a)$.

Enveloping cone of the given sphere from a point $(x_1,y_1,z_1)$ is:

$(x^2+y^2+z^2- a^2)(x_1^2+y_1^2+z_1^2- a^2) = (xx_1+yy_1+zz_1- a^2)^2$

This meets the plane $z = a$ in the conic:

$(x^2+y^2)(x_1^2+y_1^2+z_1^2- a^2) = (xx_1+yy_1+az_1- a^2)^2$

So how do we find the focal point of this conic now?