A circle in the xy -plane has a diameter with endpoints 2 , 4 and 2 , 14 . An

Question

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 ?

Accepted Answer

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&prime;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&rsquo;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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