Proper pencil through a line

Statement

Write the equation of the proper pencil of planes through the line $r:\begin{cases}x+y=1\\z=0\end{cases}$ and find the plane of the pencil that contains $P(0;0;5)$.

Solution

The line $r$ is the intersection of the two planes $\pi_1: x+y-1=0$ and $\pi_2: z=0$. The proper pencil with axis $r$ is $(x+y-1)+\lambda z = 0.$ Imposing passage through $P(0;0;5)$: $(0+0-1)+\lambda\cdot 5 = 0 \iff \lambda = \frac15.$ The required plane is $(x+y-1)+\tfrac15 z=0$, that is $\boxed{5x+5y+z-5=0}.$

Links

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