Tangent plane to double sided cone

Given $\displaystyle z^2=x^2+y^2$, show that every tangent plane goes through the origin.

We know that the normal to the tangent plane is given by the grad(F) and can spilt the function into an upper and lower half to find that

$\displaystyle f(x,y,z)=\sqrt{x^2+y^2}\Rightarrow grad(F)=(\frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x ^2+y^2}},0)$. But when we evaluate at a point $\displaystyle P(x_0,y_0,z_0)$ we get:

$\displaystyle \frac{x_0}{\sqrt{x_0^2+y_0^2}}(x-x_0)+\frac{y_0}{\sqrt{x_0^2+y_0^2}}(y-y_0)=0$. From what we can see this need not be zero at $\displaystyle (0,0,0)$