Question

What is the maximum number of solutions each of the following systems could have?

Two distinct concentric circles:

Two distinct parabolas:

A line and a circle:

A parabola and a circle:

Answer 1

Answers:

- Two distinct concentric circles: 0.
(Concentric circles with different radii have no common points.)

- Two distinct parabolas: 4.
(Two conics can meet in at most 4 points — Bezout’s theorem / degree-counting.)

- A line and a circle: 2.
(Substitute the line into the circle gives a quadratic; at most two real intersections.)

- A parabola and a circle: 4.
(Substituting the parabola into the circle yields a quartic; at most four real intersections — again consistent with Bezout.)

Each “maximum” is attainable in suitable configurations (except the concentric circles case, where distinct circles truly have zero intersections).