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Geometry and Trigonometry / Circles Difficulty: Hard

In the xy-plane, the graph of $2 x squared, minus 6 x, plus 2 y squared, plus 2 y, equals 45$ is a circle. What is the radius of the circle?

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Explanation

Choice A is correct. One way to find the radius of the circle is to rewrite the given equation in standard form, $open parenthesis, x minus h, close parenthesis, squared, plus, open parenthesis, y minus k, close parenthesis, squared, equals r squared$ , where $the ordered pair h comma k$ is the center of the circle and the radius of the circle is r. To do this, divide the original equation, $2 x squared, minus 6 x, plus 2 y squared, plus 2 y, equals 45$ , by 2 to make the leading coefficients of $x squared$ and $y squared$ each equal to 1: $as follows: x squared, minus 3 x, plus y squared, plus y, equals 22 point 5$ . Then complete the square to put the equation in standard form. To do so, first rewrite $x squared, minus 3 x, plus y squared, plus y, equals 22 point 5$ as $open parenthesis, x squared, minus 3 x, plus 2 point 2 5, close parenthesis, minus 2 point 2 5, plus, open parenthesis, y squared, plus y, plus 0 point 2 5, close parenthesis, minus 0 point 2 5, equals 22 point 5$ . Second, add 2.25 and 0.25 to both sides of the equation: $open parenthesis, x squared, minus 3 x, plus 2 point 2 5, close parenthesis, plus, open parenthesis, y squared, plus y, plus 0 point 2 5, close parenthesis, equals 25$ . Since $x squared, minus 3 x, plus 2 point 2 5, equals, open parenthesis, x minus 1 point 5, close parenthesis, squared$ , $y squared, plus y, plus 0 point 2 5, equals, open parenthesis, y plus 0 point 5, close parenthesis, squared$ , and $25 equals 5 squared$ , it follows that $open parenthesis, x minus 1 point 5, close parenthesis, squared, plus, open parenthesis, y plus 0 point 5, close parenthesis, squared, equals 5 squared$ . Therefore, the radius of the circle is 5.

Choices B, C, and D are incorrect and may be the result of errors in manipulating the equation or of a misconception about the standard form of the equation of a circle in the xy-plane.