A variable sphere passes through the points $(0,0,\pm c)$ and cuts the lines $y=x\tan\alpha$, $z=c$; $y=-x\tan\alpha$, $z=-c$ in the points $P,P'$. If $PP'$ has constant length $2a$ show that the center of the sphere lies on the line $z=0$, $x^2+y^2=(a^2-c^2)\csc^22\alpha$.
The variable sphere is passing through $(0,0,c),(0,0,-c),(x_1,x_1\tan\alpha,c),(x_1,-x_1\tan\alpha,-c)$.
Let the center of the sphere be $(X,Y,Z)$.
So $X^2+Y^2+(Z-c)^2=X^2+Y^2+(Z+c)^2=(X-x_1)^2+(Y-x_1\tan\alpha)^2+(Z-c)^2=(X-x_1)^2+(Y+x_1\tan\alpha)^2+(Z+c)^2$
Solving $X^2+Y^2+(Z-c)^2=X^2+Y^2+(Z+c)^2$ I get $Z=0$ but I am not able to get $X^2+Y^2=(a^2-c^2)\csc^22\alpha$.
Please help.