Improper pencil parallel to a plane

Statement

Determine the improper pencil of planes parallel to $\pi\colon 2x-3y+z-7=0$ and find the plane of the pencil passing through $Q(1;1;1)$.

Solution

The planes parallel to $\pi$ have the same normal vector $(2;-3;1)$; only the constant term varies: $2x-3y+z+k=0.$ Imposing passage through $Q(1;1;1)$: $2-3+1+k=0 \iff k=0.$ The required plane is $\boxed{2x-3y+z=0}.$

Links

Topics: Analytic geometry in space
Concepts: Pencil of planes · Plane
Methods: Improper pencil of planes
Skills: Analytic geometry · Using formulae
Exercise types: Geometric problem