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Geometry and Trigonometry / Circles Difficulty: Medium

A circle in the xy-plane has its center at $left parenthesis negative 4 comma 5 right parenthesis$ and the point $left parenthesis negative 8 comma 8 right parenthesis$ lies on the circle. Which equation represents this circle?

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Explanation

Choice D is correct. A circle in the xy-plane can be represented by an equation of the form ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ , where $(h, k)$ is the center of the circle and $r$ is the length of a radius of the circle. It's given that the circle has its center at $left parenthesis negative 4 comma 5 right parenthesis$ . Therefore, $h = - 4$ and $k equals 5$ . Substituting $negative 4$ for $h$ and $5$ for $k$ in the equation ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ yields ${(x - (- 4))}^{2} + {(y - 5)}^{2} = r^{2}$ , or ${(x + 4)}^{2} + {(y - 5)}^{2} = r^{2}$ . It's also given that the point $left parenthesis negative 8 comma 8 right parenthesis$ lies on the circle. Substituting $negative 8$ for $x$ and $8$ for $y$ in the equation ${(x + 4)}^{2} + {(y - 5)}^{2} = r^{2}$ yields ${(- 8 + 4)}^{2} + {(8 - 5)}^{2} = r^{2}$ , or ${(- 4)}^{2} + {(3)}^{2} = r^{2}$ , which is equivalent to $16 + 9 = r^{2}$ , or $25 equals r squared$ . Substituting $25$ for $r^{2}$ in the equation ${(x + 4)}^{2} + {(y - 5)}^{2} = r^{2}$ yields ${(x + 4)}^{2} + {(y - 5)}^{2} = 25$ . Thus, the equation ${(x + 4)}^{2} + {(y - 5)}^{2} = 25$ represents the circle.

Choice A is incorrect. The circle represented by this equation has its center at $left parenthesis 4 comma negative 5 right parenthesis$ , not $left parenthesis negative 4 comma 5 right parenthesis$ , and the point $left parenthesis negative 8 comma 8 right parenthesis$ doesn't lie on the circle.

Choice B is incorrect. The point $left parenthesis negative 8 comma 8 right parenthesis$ doesn't lie on the circle represented by this equation.

Choice C is incorrect. The circle represented by this equation has its center at $left parenthesis 4 comma negative 5 right parenthesis$ , not $left parenthesis negative 4 comma 5 right parenthesis$ , and the point $left parenthesis negative 8 comma 8 right parenthesis$ doesn't lie on the circle.