Determining the parameter

The typical problem on pencils is: given a pencil $f_k$, find the value (or values) of $k$ for which the corresponding line satisfies a certain geometric condition. This section presents the general strategy and illustrates it with four progressively richer examples: passing through a point, fixed distance, tangency to a parabola and secant/tangent/external classification. A summary of the method closes it.

• The strategy for determining the parameter
• Example — Passing through a point
• Example — Fixed distance
• Example — Tangency to a parabola
• Example — Secant, tangent, external
• General method for pencil problems