{
ParametricPlot3D[{x,y,z}/.Solve[{x^2+y^2==z^2,z-z0-y Tan[θ]==0}/.{z0->1/2,θ->Pi/3},
{y,z}]//Evaluate,{x,-2,2}],
ParametricPlot3D[{Sqrt[1/8]Tan[t],1/4 (-Sqrt[3]+Sec[t]),1/4 (-1+Sqrt[3] Sec[t])},{t,0,2Pi}]
}
{
ParametricPlot3D[{x,y,z}/.Solve[{x^2+y^2==z^2,z-z0-y Tan[θ]==0}/.{z0->1/3,θ->Pi/6},{y,z}]//Evaluate,{x,-2,2}],
ParametricPlot3D[{Sqrt[1/6]Cos[t],1/6 (Sqrt[3]-3 Sin[t]),1/6 (3-Sqrt[3] Sin[t])},{t,0,2Pi}]
}