A line with a single common point with the parabola without being tangent

Text

Draw a parabola and a line such that the line has a single point (NOT two coincident points) in common with the parabola.

Solution

A line and a parabola can have $2$, $1$ or $0$ common points. The ”single point” case occurs in two ways:

• the line is tangent: the system gives a quadratic equation with two coincident solutions (zero discriminant) — but this case is excluded by the text;
• the line is parallel to the axis of the parabola (for a parabola with vertical axis, a vertical line $x=k$): the system reduces to a first-degree equation, with a single solution, not a double one.

Example: parabola $y=x^2$ and vertical line $x=1$. Substituting gives the unique point $(1;1)$; there is no second point because to each $x$ the parabola associates only one $y$.

$\boxed{\text{A line parallel to the axis (e.g. }x=1\text{) meets the parabola at a single point, without being tangent.}}$