Complete the square in the equation for both x and y to locate the center. Then, find the midpoint of AB to see if they are one and the same. The center of the circle would be the midpoint of the diameter.

Center is at (1, 2)

Midpoint of AB =

Substitute (4, 4) into your original equation to see if it makes a true statement

??

Angle ACB is an inscribed angle in a semicircle, therefore 90 degrees.

See Earboth's post.

Define a linear equation for the line passing through D(5, 0) and F(0, 2). Then solve the system which includes this linear equation and the equaton of the circle. You should get two distinct points of intersection. Try it.