A line segment is a diameter of a circle. Determine the radius of the circle and if...

The coordinates of A are (1, 2) and B are (5, 0). The diameter AB is $\sqrt{(5-1)^2 + (0-2)^2} = \sqrt{4^2 + (-2)^2} = \sqrt{16+4} = \sqrt{20} = 2\sqrt{5}$. The radius is half the diameter, so $\frac{2\sqrt{5}}{2} = \sqrt{5}$. The center C is the midpoint of AB, which is $(\frac{1+5}{2}, \frac{2+0}{2}) = (3, 1)$. The equation of the circle is $(x-3)^2 + (y-1)^2 = (\sqrt{5})^2 = 5$. For point E(2, 4), $(2-3)^2 + (4-1)^2 = (-1)^2 + 3^2 = 1 + 9 = 10$. Since $10 \neq 5$, point E does not lie on the circle.