General method for pencil problems

All pencil problems follow the same five-step scheme.

In summary — General method for pencil problems

1. Recognise whether the pencil is proper or improper by collecting the parameter; identify the two generators.
2. Translate the geometric condition into an equation or inequality in $k$.
3. Solve for $k$.
4. Check the generator excluded from the parametrisation.
5. Write the solution lines by substituting the values of $k$ found.

Links

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