Find the cylindrical surface with directrix y^2=4x, z=0 and…

Find the cylindrical surface with directrix $y^2=4x,\ z=0$ and generators parallel to $\vec p=\{1,2,3\}$.

Explanation of Method

The cylinder through a parabola in xOy with generators parallel to a given vector.

The generator through $M(x,y,z)$ is $(x+t,\ y+2t,\ z+3t)$. It meets $z=0$ at $t=-\tfrac z3$, at the point

\[ \left(x-\tfrac z3,\ y-\tfrac{2z}3,\ 0\right). \]

\[ \left(y-\tfrac{2z}3\right)^2=4\left(x-\tfrac z3\right)\iff (3y-2z)^2=12(3x-z). \]

\[9y^2-12yz+4z^2-36x+12z=0\]

Check: $(1,2,0)+\{1,2,3\}=(2,4,3)$ gives $144-144+36-72+36=0$. $\checkmark$

$\vec p$ $y^2=4x$

Generators (teal) through the parabola $y^2=4x,\ z=0$ (red), all parallel to $\vec p=\{1,2,3\}$.

\[\boxed{\,9y^2-12yz+4z^2-36x+12z=0\,}\]