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.

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.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Equations of Circles", "Vertex of a Quadratic Function", "Graphical Intersection of Systems" ], "new_concepts": [], "current_concepts": [ "Equations of Circles", "Vertex of a Quadratic Function", "Graphical Intersection of Systems" ] } ## Determine the circle's radius $$ \begin{aligned} &\text{Center of circle: } (0,0)\ &\text{Vertex of parabola: } (0,9)\ &\text{Since the circle intersects the parabola at its vertex, the point } (0,9) \text{ lies on the circle.}\ &\text{Radius } r = \sqrt{(0-0)^2 + (9-0)^2} = 9 \end{aligned} $$ ## Analyze the intersection of the circle and the upward-opening parabola $$ \begin{aligned} &\text{Circle equation: } x^2 + y^2 = 81\ &\text{Parabola equation: } y = ax^2 + 9 \quad (a > 0)\ &\text{Substitute } x^2 = \frac{y-9}{a} \text{ into the circle equation:}\ &\frac{y-9}{a} + y^2 = 81 \implies ay^2 + y - (9 + 81a) = 0\ &\text{Since } y \ge 9 \text{ for the upward-opening parabola, we check if other solutions exist:}\ &\text{For } y > 9 \implies y^2 > 81 \implies x^2 + y^2 > 81 \text{ (no intersection points on the circle above } y = 9\text{).}\ &\text{Thus, } y = 9 \text{ is the only possible } y\text{-coordinate, yielding exactly one intersection point: } (0,9). \end{aligned} $$ ## Evaluate the given statements $$ \begin{aligned} &\text{1. "The maximum number of solutions is one" } \implies \text{True (the only intersection is } (0,9)\text{).}\ &\text{2. "The maximum number of solutions is three" } \implies \text{False.}\ &\text{3. "The circle has a radius equal to 3" } \implies \text{False (radius is 9).}\ &\text{4. "The circle has a radius less than 9" } \implies \text{False (radius is 9).} \end{aligned} $$

(A) The maximum number of solutions is one

(B) The maximum number of solutions is three

(C) The circle has a radius equal to 3

(D) The circle has a radius less than 9

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QUESTION IMAGE

Question

Explanation:

Determine the circle's radius

Analyze the intersection of the circle and the upward-opening parabola

Evaluate the given statements

Answer: