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Geometry and Trigonometry / Circles Difficulty: Hard

$x squared, plus 20 x, plus y squared, plus 16 y, equals negative 20$

The equation above defines a circle in the xy-plane. What are the coordinates of the center of the circle?

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Explanation

Choice B is correct. The standard equation of a circle in the xy-plane is of the form $open parenthesis, x minus h, close parenthesis, squared, plus, open parenthesis, y minus k, close parenthesis, squared, equals r squared$ , where $the ordered pair h comma k$ are the coordinates of the center of the circle and r is the radius. The given equation can be rewritten in standard form by completing the squares. So the sum of the first two terms, $x squared, plus 20 x$ , needs a 100 to complete the square, and the sum of the second two terms, $y squared, plus 16 y$ , needs a 64 to complete the square. Adding 100 and 64 to both sides of the given equation yields $open parenthesis, x squared, plus 20 x, plus 100, close parenthesis, plus, open parenthesis, y squared, plus 16 y, plus 64, close parenthesis, equals negative 20, plus 100, plus 64$ , which is equivalent to $open parenthesis, x plus 10, close parenthesis, squared, plus, open parenthesis, y plus 8, close parenthesis, squared, equals 144$ . Therefore, the coordinates of the center of the circle are $negative 10 comma negative 8$ .

Choices A, C, and D are incorrect and may result from computational errors made when attempting to complete the squares or when identifying the coordinates of the center.