Write the equation of the plane that is perpendicular to…

Write the equation of the plane that is perpendicular to $5x-y+3z-2=0$ and intersects it along a line lying in the $xOy$ plane.

Step 1: The common line

The common line lies in the given plane and in $z=0$: \[\ell:\ 5x-y-2=0,\ z=0,\qquad \vec d=\vec k\times\vec n_1=\{1,5,0\}.\]

Step 2: Pencil through $\ell$

The pencil is $5x-y-2+\lambda z=0$. Perpendicularity to the given plane: \[\{5,-1,\lambda\}\cdot\{5,-1,3\}=26+3\lambda=0\ \Longrightarrow\ \lambda=-\tfrac{26}{3} .\]

\[15x-3y-26z-6=0\]

Check: $P\left(\tfrac25,0,0\right)$ and $Q(0,-2,0)$ satisfy it, $\{15,-3,-26\}\cdot\{5,-1,3\}=75+3-78=0$, and for $z=0$ the equation reduces to $5x-y-2=0$.