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

Explanation:

Step1: Analyze the geometric shapes

A circle is a closed - curve defined by the equation \((x - a)^{2}+(y - b)^{2}=r^{2}\) (where \((a,b)\) is the center and \(r\) is the radius), and a parabola is defined by an equation of the form \(y = Ax^{2}+Bx + C\) (for a parabola opening up or down) or \(x=Ay^{2}+By + C\) (for a parabola opening left or right).

Step2: Consider the number of intersection points

• 0 intersection points: If the circle and the parabola are completely separate from each other. For example, a parabola \(y=x^{2}+10\) (vertex at \((0,10)\)) and a circle \((x - 0)^{2}+(y - 0)^{2}=1\) (center at \((0,0)\) and radius \(r = 1\)) will not intersect.
• 1 intersection point: When the circle is tangent to the parabola. For instance, a circle \((x - 0)^{2}+(y - k)^{2}=r^{2}\) and a parabola \(y=x^{2}\) can be tangent at one point.
• 2 intersection points: A common case. For example, a circle \((x - 0)^{2}+(y - 0)^{2}=4\) and a parabola \(y=x^{2}- 2\) will intersect at two points.
• 3 intersection points: We can think of a circle that intersects a parabola at three points. For example, consider a parabola \(y=x^{2}\) and a circle that is "sliced" by the parabola in three places.
• 4 intersection points: A circle \((x - 0)^{2}+(y - 0)^{2}=25\) and a parabola \(y=x^{2}-5\) will intersect at four points.

Since a circle is a second - degree curve and a parabola is a second - degree curve, the maximum number of intersection points of two second - degree curves is given by the formula \(n_1\times n_2\) (where \(n_1\) and \(n_2\) are the degrees of the curves). Here \(n_1=n_2 = 2\), so the maximum number of intersection points is \(4\). A circle and a parabola cannot intersect at \(5\) points.

Answer:

0, 1, 2, 3, 4