Question

What’s the maximum amount of solutions a system containing a circle and parabola can have?

Answer 1

Answer: The maximum number of solutions is four.

Step-by-step explanation: To find the number of solutions of a system containing a circle and a parabola, we need to find the points of intersection between the two curves. The points of intersection are the solutions of the system of equations that represent the circle and the parabola. For example, if the circle has the equation \ce {x^2 + y^2 = r^2} x^2 + y^2 = r^2 and the parabola has the equation \ce {y = ax^2 + bx + c} y = ax^2 + bx + c, then we can substitute \ce {y} y from the second equation into the first equation and get a quadratic equation in \ce {x} x:

\ce {x^2 + (ax^2 + bx + c)^2 = r^2} x^2 + (ax^2 + bx + c)^2 = r^2

This equation can have at most two real roots for \ce {x} x, which correspond to at most two points of intersection. However, this is not the maximum number of solutions possible, because we can also substitute \ce {x} x from the second equation into the first equation and get another quadratic equation in \ce {y} y:

\ce {(y - c - bx)^2 / a^2 + y^2 = r^2} (y - c - bx)^2 / a^2 + y^2 = r^2

This equation can also have at most two real roots for \ce {y} y, which correspond to at most two more points of intersection. Therefore, the maximum number of solutions is four, when both quadratic equations have two real roots each. This happens when the circle and the parabola intersect in four distinct points, as shown in this example1:

\ce {x^2 + y^2 = 25} x^2 + y^2 = 25

\ce {y = x^2 - 4x + 4} y = x^2 - 4x + 4

The points of intersection are (5, 0), (-5, 0), (3, 4), and (-3, 4).

Hope this helps, and have a great day! =)

Answer 2

The maximum number of solutions for a system containing a circle and a parabola is four, as these shapes can intersect at a maximum of four points on a graph.

Step-by-step explanation:

The maximum amount of solutions that a system containing a circle and parabola can have is four. This conclusion is derived from the fact that a circle, which is a type of ellipse, and a parabola are both conic sections. When graphing both shapes on a coordinate plane, the number of intersecting points reflects the possible solutions to the system formed by the equations of the circle and parabola. Since a parabola has a single curved path and a circle is round, they can touch at a maximum of four points. However, the actual number of intersecting points may vary from zero (if they do not touch at all) to four, depending on their relative position and size.