A circle in the xy-plane has a diameter with endpoints (2, 4) and (2, 14). An equation of this circle is (x − 2) 2 + (y − 9) 2 = r 2 , where r is a positive constant. What is the value of r?

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The correct answer is 5. The standard form of an equation of a circle in the xy-plane is (x − h) 2 + (y − k) 2 = r 2 , where h, k, and r are constants, the coordinates of the center of the circle are (h, k), and the length of the radius of the circle is r. It’s given that an equation of the circle is (x − 2) 2 + (y − 9) 2 = r 2 . Therefore, the center of this circle is (2, 9). It’s given that the endpoints of a diameter of the circle are (2, 4) and (2, 14). The length of the radius is the distance from the center of the circle to an endpoint of a diameter of the circle, which can be found using the distance formula, √((x 1 − x 2 ) 2 + (y 1 − y 2 ) 2 ). Substituting the center of the circle (2, 9) and one endpoint of the diameter (2, 4) in this formula gives a distance of √((2 − 2) 2 + (9 − 4) 2 ), or √(0 2 + 5 2 ), which is equivalent to 5. Since the distance from the center of the circle to an endpoint of a diameter is 5, the value of r is 5.

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