Example — Passing through a point

The simplest condition: requiring that the line of the pencil pass through an assigned point. It suffices to substitute the coordinates of the point into the equation of the pencil and solve for $k$.

Example — Passing through a point

In the pencil $f_k:\ x - ky + k + 2 = 0$, find the line that passes through $Q(1;0)$.

Substitution: $1 - k\cdot 0 + k + 2 = 0 \implies 3 + k = 0 \implies k = -3$.

Required line: $x - (-3)y + (-3) + 2 = 0$, that is $x + 3y - 1 = 0$.

Links

Topics: Pencils of lines
Concepts: Pencil of lines · Proper pencil · Pencil parameter
Functions: Line
Skills: Analytic geometry · Solving equations