In the xy -plane, the graph of the equation above is a circle. Point P is on the circle and

Question

In the xy -plane, the graph of the equation above is a circle. Point P is on the circle and has coordinates . If is a diameter of the circle, what are the coordinates of point Q ?

Accepted Answer

Choice A is correct. The standard form for the equation of a circle is , where 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 . Let represent the coordinates of point Q . Since point P and point Q are the endpoints of a diameter of the circle, the center lies on the diameter, halfway between P and Q . Therefore, the following relationships hold: and . Solving the equations for and , respectively, yields and . Therefore, the coordinates of point Q are . Alternate approach: Since point P on the circle and the center of the circle have the same y -coordinate, it follows that the radius of the circle is . In addition, the opposite end of the diameter must have the same y -coordinate as P and be 4 units away from the center. Hence, the coordinates of point Q must be . Choices B and D are incorrect because the points given in these choices lie on a diameter that is perpendicular to the diameter . If either of these points were point Q , then would not be the diameter of the circle. Choice C is incorrect because is the center of the circle and does not lie on the circle.

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