Answer
a) $(4,5)$
b) $r=\sqrt 2$
c) $(x-4)^2+(y-5)^2=2$
Work Step by Step
a) The given co-ordinates are $(3, 6)$ and $(5, 4)$. It is given that the line segment that intersects the circle at the given co-ordinates passes through the the center of the circle, meaning the line segment is a diameter of the circle.
The center of the circle is the midpoint of the diameter:
$$((3+5)/2, (6+4)/2)=(4,5).$$
b) The radius is half the distance between the points $(3,6)$ and $(5,4)$:
$$r=\frac{1}{2}\sqrt{(5-3)^2+(4-6)^2}=\frac{1}{2}\sqrt{8}=\frac{1}{2}\cdot 2\sqrt 2=\sqrt 2$$ $$r=\sqrt 2$$
c) The equation of the circle is:
$$(x-4)^2+(y-5)^2=2.$$
