Find the center and radius of the circle passing through the points A (-1; 5) B (-2; -2) C (5; 5).

Let’s write down the general form of equations of circles:

(x – a) ^ 2 + (y – b) ^ 2 = R ^ 2.

Substitute the coordinates of the points we know into the circle formula:

(-1 – a) ^ 2 + (5 – b) ^ 2 = R ^ 2;

(-2 – a) ^ 2 + (-2 – b) ^ 2 = R ^ 2;

(5 – a) ^ 2 + (5 – b) ^ 2 = R ^ 2;

Let’s equate the left sides of the first and third equations:

(1 + a) ^ 2 + (5 – b) ^ 2 = (5 – a) ^ 2 + (5 – b) ^ 2;

a ^ 2 + 2 * a + 1 = a ^ 2 – 10 * a + 25;

12 * a = 24;

a = 2.

Now we equate the first and second equations:

(1 + a) ^ 2 + (5 – b) ^ 2 = (2 + a) ^ 2 + (2 + b) ^ 2;

9 + b ^ 2 – 10 * b + 25 = 16 + b ^ 2 + 4 * b + 4;

14 * b = 14;

b = 1.

We get:

R ^ 2 = 3 ^ 2 + 4 ^ 2 = 25;

R = 5.

The center of the circle is (2; 1).