All tangent planes to a surface of revolution (S) at points M of A (intersection of (S) with the plane of equation: z=a) pass by a fixed point.

Question

I'm faced with this problem. The complete problem statement

In part 6, I'm asked to show that all the tangent planes to the surface of revolution (S) at points M of A (which is the intersection of (S) with the plane of equation: z = a) pass by a fixed point to be determined.

Here is what I've tried:

In part 3, I've found that the cartesian equation of (S) is: $x^2z^2 + y^2z^2 = 1$ then in part 6 I tried to find the equation of the tangent plane to (S) at point $M=(x_0,y_0,z_0)$ which is: ${\textbf{MX}.{\nabla{F}}}=0$ where $X=(x,y,a)$ but I don't know if this is true.