Line–parabola intersections

Where do a straight line and a parabola meet? You put the two equations into a system and obtain a second-degree equation: the discriminant tells us whether there are $0$, $1$ or $2$ intersections. If $\Delta<0$ the line is external, if $\Delta=0$ it is tangent, if $\Delta>0$ it is secant.

Example

Given the parabola $y=\frac{x^2}{4}+x+1$ and the line $r:\ y=\frac12 x$, determine whether $r$ is tangent, secant or external.

Putting them into a system: $\frac{x^2}{4}+x+1=\frac12 x \implies \frac{x^2}{4}+\frac{x}{2}+1=0 \implies x^2+2x+4=0.$ The discriminant is $\Delta=4-16=-12<0$: the line is external to the parabola.

A generic line parallel to $r$ has equation $y=\frac12 x+q$, with $q$ a parameter.

Links

Topics: Parabola
Concepts: Discriminant · Line–parabola intersection · System
Functions: Parabola
Skills: Analytic geometry · Solving systems