Answer
a) $(5,10)$
b) $r=\sqrt 5$
c) $(x-5)^2+(y-10)^2=5$
Work Step by Step
a) The given co-ordinates are $(3, 9)$ and $(7, 11)$. 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:
$$((7+3)/2, (11+9)/2)=(5,10).$$
b) The radius is half the distance between the points $(3,9)$ and $(7,11)$:
$$r=\frac{1}{2}\sqrt{(7-3)^2+(11-9)^2}=\frac{1}{2}\sqrt{20}=\frac{1}{2}\cdot 2\sqrt 5=\sqrt 5$$ $$r=\sqrt 5$$
c) The equation of the circle is:
$$(x-5)^2+(y-10)^2=5.$$
