A circle passes through (-1,3) and (5,11) with radius 5. How do I go about finding its center?

Question

A circle passes through (-1,3) and (5,11) with radius 5. How do I go about finding its center?

I tried writing the equations but it became too complicated. Is there a simple way?

I tried $x²+y²+2gx+2fy+c= 0$ Then
$c= g²+f²-r²$ but it ended up in complicated cases

Answer 1

Note that the distance between the two given points is $\sqrt{(11-3)^2+(5-(-1))^2}=10$. This means that for a circle of radius $5$ to pass through them, it must have the segment between those two points as its diameter, and so its centre must be the midpoint, or $(2,7)$.

Answer 2

Observe that the distance between two points is the diameter so the center is the midpoint.

Answer 3

If the two points did not happen two be opposite each other on the circle, a more general way to solve this problem is to solve the system of equations: $$(x_1-r_x)^2+(y_1-r_y)^2 = r^2$$ $$(x_2-r_x)^2+(y_2-r_y)^2 = r^2,$$ where the radius $r$ and two points $(x_1,y_1)$ and $(x_2,y_2)$ are given.