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Geometry and Trigonometry Difficulty: Hard

$Open parenthesis, x minus 6, close parenthesis, squared, plus, open parenthesis, y plus 5, close parenthesis, squared, equals 16$

In the xy-plane, the graph of the equation above is a circle. Point P is on the circle and has coordinates $10 comma negative 5$ . If $line segment P Q$ is a diameter of the circle, what are the coordinates of point Q ?

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Explanation

Choice A is correct. The standard form for the equation of a circle is $open parenthesis, x minus h, close parenthesis, squared, plus, open parenthesis, y minus k, close parenthesis, squared, equals r squared$ , where $the ordered pair h comma k$ are the coordinates of the center and r is the length of the radius. According to the given equation, the center of the circle is $the point with coordinates 6 comma negative 5$ . Let $x sub 1 comma y sub 1$ represent the coordinates of point Q. Since point P $10 comma negative 5$ and point Q $x sub 1 comma y sub 1$ are the endpoints of a diameter of the circle, the center $with coordinates 6 comma negative 5$ lies on the diameter, halfway between P and Q. Therefore, the following relationships hold: $the fraction with numerator x sub 1, plus 10, and denominator 2, equals 6$ and $the fraction with numerator y sub 1, plus negative 5, and denominator 2, equals negative 5$ . Solving the equations for $x sub 1$ and $y sub 1$ , respectively, yields $x sub 1, equals 2$ and $y sub 1, equals negative 5$ . Therefore, the coordinates of point Q are $2 comma negative 5$ .

Alternate approach: Since point P $10 comma negative 5$ on the circle and the center of the circle $6 comma negative 5$ have the same y-coordinate, it follows that the radius of the circle is $10 minus 6, equals 4$ . In addition, the opposite end of the diameter $P Q$ must have the same y-coordinate as P and be 4 units away from the center. Hence, the coordinates of point Q must be $2 comma negative 5$ .

Choices B and D are incorrect because the points given in these choices lie on a diameter that is perpendicular to the diameter $P Q$ . If either of these points were point Q, then $line segment P Q$ would not be the diameter of the circle. Choice C is incorrect because $the point with coordinates 6 comma negative 5$ is the center of the circle and does not lie on the circle.