Solution (source code)

= Solution
{c}

Every point of the indicated ray has the form $w_0=(r,2r,\sqrt5\,r)$ with $r>0$; the apex is excluded. Substitution into the preceding <tangent space> formula gives
$$
\boxed{T_{w_0}M=\{(\xi,\eta,\zeta):\xi+2\eta-\sqrt5\,\zeta=0\},}
$$
independent of $r$. The affine tangent plane has the same equation, since the ray lies in this plane. Thus both interpretations of the requested tangent plane are constant along this generator of the <cone>.