Past exam of the options I course of the University of Oxford / 2011 / F / 1 / b / ii / Solution

The tangent space to a regular level set is the kernel of its derivative. Writing $z_0=\sqrt{x_0^2+y_0^2}$ givesIf an affine tangent plane through $v_0$ is intended, its equation is $x_0(x-x_0)+y_0(y-y_0)-z_0(z-z_0)=0$. Since $x_0^2+y_0^2=z_0^2$, this simplifies to the same plane $x_{0}x+y_{0}y-z_{0}z=0$, which passes through the apex.