Problem:
The circle having and as the endpoints of a diameter intersects the -axis at a second point. What is the -coordinate of this point?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The center of the circle is the midpoint of the diameter, which is , and the radius is . Therefore the equation of the circle is . If , then , so or . The circle intersects the -axis at .
OR
Any diameter of a circle is a line of symmetry. Because the line goes through the center of the circle, , it contains a diameter. The reflection of in this line is . Alternatively, can be reflected in the line , resulting in the same point.
The problems on this page are the property of the MAA's American Mathematics Competitions