Question

the center of a circle is at the origin on a coordinate grid. the vertex of a parabola that opens upward is at (0, 9). if the circle intersects the parabola at the parabolas vertex, which statement must be true? the maximum number of solutions is one. the maximum number of solutions is three. the circle has a radius equal to 3. the circle has a radius less than 9.

Explanation:

Step1: Analyze the intersection

A circle centered at the origin and a parabola with vertex at (0, 9) intersect at the vertex of the parabola. The equation of a circle centered at the origin is $x^{2}+y^{2}=r^{2}$ and for an upward - opening parabola with vertex at (0, 9) its general form is $y = ax^{2}+9$ ($a>0$). Since they intersect at (0, 9), substituting $x = 0$ and $y = 9$ into the circle equation gives $0^{2}+9^{2}=r^{2}$, so $r = 9$ is a possibility when they just touch at the vertex. If the circle is larger, it will intersect the parabola at more points.

Step2: Evaluate each option

• Option 1: Since the parabola opens upward and the circle is centered at the origin, if they intersect at the vertex, there could be more intersection points if the circle is larger. So the maximum number of solutions is not necessarily one.
• Option 2: It is possible for a circle and a parabola to have 0, 1, 2, 3, or 4 intersection points. But just because they intersect at the vertex doesn't mean the maximum is three.
• Option 3: There is no information to suggest that the radius is 3.
• Option 4: If the circle has a radius less than 9, it will only intersect the parabola at the vertex (if it intersects at all). If $r = 9$, they touch at the vertex, and if $r>9$, they will intersect at more than one point.

Answer:

The circle has a radius less than 9.