Example — Secant, tangent, external

By studying the sign of the discriminant as a function of $k$ one classifies, all together, the lines of the pencil according to their position with respect to the curve: secant, tangent or external.

Example — Secant, tangent, external

In the proper pencil $f_k:\ k(3x-2y+1) + (6x-4y+1) = 0$ and given the parabola $p: y = 2x^2 + 7$, determine the values of $k$ for which the line of the pencil is (a) tangent, (b) secant, (c) external to $p$.

System: $\begin{cases} y = 2x^2 + 7 \\ k(3x-2y+1) + 6x -4y +1 = 0\end{cases}$ Substituting $y$ into the second equation we obtain a second-degree polynomial in $x$ with coefficients depending on $k$; its discriminant $\Delta(k)$ is, in turn, a polynomial in $k$. We have:

• $\Delta(k) > 0$: two distinct intersections $\Rightarrow$ secant;
• $\Delta(k) = 0$: a double intersection $\Rightarrow$ tangent;
• $\Delta(k) < 0$: no intersection $\Rightarrow$ external.

The study of the sign of $\Delta(k)$ gives the three subsets of values of $k$ corresponding to the three conditions. At the end, as always, one must check whether the generator excluded from the pencil (the one that would formally be obtained with $k\to\infty$) is also a solution.

Links

Topics: Pencils of lines
Concepts: Discriminant · Pencil of lines · Proper pencil · Pencil parameter · Line · External line · Secant line · Tangency
Functions: Parabola · Line
Skills: Analytic geometry · Reasoning by cases · Solving systems