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Geometry and Trigonometry Difficulty: Medium

In the xy-plane, the graph of the equation ${(x - 3)}^{2} + {(y - 5)}^{2} = 9$ is a circle. The point $left parenthesis 6 comma c right parenthesis$ , where $c$ is a constant, lies on this circle. What is the value of $c$ ?

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Explanation

The correct answer is $5$ . It's given that in the xy-plane, the graph of the equation ${(x - 3)}^{2} + {(y - 5)}^{2} = 9$ is a circle. It’s also given that the point $left parenthesis 6 comma c right parenthesis$ , where $c$ is a constant, lies on this circle. It follows that the ordered pair $left parenthesis 6 comma c right parenthesis$ makes the equation ${(x - 3)}^{2} + {(y - 5)}^{2} = 9$ true. Substituting $6$ for $x$ and $c$ for $y$ in this equation yields ${(6 - 3)}^{2} + {(c - 5)}^{2} = 9$ , or $9 + {(c - 5)}^{2} = 9$ . Subtracting $9$ from each side of this equation yields ${(c - 5)}^{2} = 0$ . It follows that the value of $c$ is $5$ .