The improper pencil

The second type of pencil arises when we fix the slope and leave the position of the line free.

Definition — Improper pencil

An improper pencil of lines is the set of all the lines of the plane parallel to a fixed direction (given by the slope $m_0$). Its equation is: $y = m_0 x + q, \qquad q\in\mathbb{R},$ where the parameter $q$ is the $y$-intercept. The improper pencil has no “centre”: all its lines are parallel to one another.

Here the parameter that runs through the pencil is the $y$-intercept $q$: as $q$ varies the line translates while always keeping the same slope $m_0$.

Links

Topics: Pencils of lines
Concepts: Slope · Pencil of lines · Improper pencil · Y-intercept · Line
Functions: Line
Methods: Line of an improper pencil
Skills: Analytic geometry · Using formulae