Question

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

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


0

1

2

3

4

5

Answer 1

0, 1, 2, 3, 4

Reason: A circle and a parabola are two conics, so by Bézout’s theorem they meet in at most 4 points (counting multiplicity). All counts 0–4 can occur with suitable positions; 5 cannot.

Examples:
- 0: circle centered at (0,−1) radius 0.2, x^2+(y+1)^2=0.04, with y=x^2 (parabola has y≥0, circle has y<0) — no intersection.
- 1: circle x^2+(y−1/4)^2=(1/4)^2 and y=x^2 meet only at (0,0) (tangency).
- 2: circle x^2+y^2=1 with y=x^2 gives two intersections (x ≈ ±0.786, y ≈0.618).
- 3: circle x^2+(y−1)^2=1 with y=x^2 meets at (0,0) and (±1,1) — three points.
- 4: circle x^2+(y−1)^2=0.81 with y=x^2 gives u^2−u+0.19=0 in u=x^2, two distinct positive u ⇒ four x-values (four intersections).