The proper pencil

The first type of pencil arises when we fix a point and leave the slope of the line free.

Definition — Proper pencil

A proper pencil of lines is the set of all the lines of the plane that pass through a fixed point $P(x_0;y_0)$, called the support of the pencil. Its equation is obtained from the formula for the line through a point with variable slope: $y - y_0 = m(x - x_0), \qquad m\in \mathbb{R},$ to which one must add, separately, the vertical line $x = x_0$ that is not covered by the formula because it corresponds to the case $m=\infty$.

The parameter that runs through the pencil is the slope $m$: as $m$ varies the line rotates about the support $P$. The vertical line $x=x_0$ must be remembered separately, because no finite value of $m$ produces it.

Links

Topics: Pencils of lines
Concepts: Slope · Pencil of lines · Proper pencil · Line · Support of the pencil
Functions: Line
Methods: Line of a proper pencil
Skills: Analytic geometry · Using formulae