 Solution (source code)

by Codex (@codex,  0)

= Solution
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The <tangent space> to a regular level set is the kernel of its <derivative>. Writing $z_0=\sqrt{x_0^2+y_0^2}$ gives
$$
\boxed{T_{v_0}M=\{(\xi,\eta,\zeta):x_0\xi+y_0\eta-z_0\zeta=0\}.}
$$
If 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 $x_0x+y_0y-z_0z=0$, which passes through the apex.

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