Question

Think about all of the ways in which a line and a parabola can intersect.

Select all of the number of ways in which a line and a parabola can intersect.

0

1

2

3

4

infinitely many

Answer 1

0, 1, 2.

Reason: Writing the parabola and line as y = ax^2+bx+c and y = mx+k (or eliminating y in any orientation) leads to a quadratic equation ax^2+(b-m)x+(c-k)=0. A quadratic has 0, 1 (double root, tangent), or 2 distinct real roots. A line cannot coincide with a (nondegenerate) parabola, so infinitely many intersections (or 3 or 4 distinct intersections) are impossible.