Asked by nb
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 parabola’s 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.
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Answer: The circle has a radius equal to 9.
Reason: If the circle (center at the origin) passes through the parabola’s vertex (0,9), its radius is the distance from (0,0) to (0,9), which is 9. (Substituting y = ax^2 + 9 into x^2 + y^2 = 81 shows x = 0 is the only real solution for a > 0, so the single intersection is at the vertex.)
Reason: If the circle (center at the origin) passes through the parabola’s vertex (0,9), its radius is the distance from (0,0) to (0,9), which is 9. (Substituting y = ax^2 + 9 into x^2 + y^2 = 81 shows x = 0 is the only real solution for a > 0, so the single intersection is at the vertex.)
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