The strategy for determining the parameter

The typical problem is: “in the pencil $f_k$, find the value (or values) of $k$ for which the corresponding line satisfies a certain condition”. The most frequent conditions are:

• passing through an assigned point $Q$;
• parallelism or perpendicularity to a given line;
• fixed distance from a point;
• tangency to a curve (parabola, circle, etc.);
• conditions of secancy or of a line being external with respect to a curve.

The strategy is always the same: translate the condition into an equation (or inequality) in the parameter $k$, solve it, check whether the generator excluded from the parametrisation is among the solutions, and combine everything.

Links

Topics: Pencils of lines
Concepts: Point–line distance · Pencil of lines · Parallelism · Pencil parameter · Perpendicularity · Line · Tangency
Functions: Line
Skills: Analytic geometry · Solving equations