Find the equation of the circle knowing that one of its diameters has the points as its ends. A (6.5) and B (-2.11).

To find the equation of a circle given the endpoints of its diameter, we first need to find the center of the circle which lies at the midpoint of the diameter.

Given points A(6, 5) and B(-2, 11), we can find the coordinates of the center C by using the midpoint formula:
Midpoint formula: M\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)

Substitute the coordinates of A and B into the midpoint formula:
C\left(\dfrac{6 + (-2)}{2}, \dfrac{5 + 11}{2}\right)
C\left(\dfrac{4}{2}, \dfrac{16}{2}\right)
C\left(2, 8\right)

So, the center of the circle is C(2, 8).

Next, we need to find the radius of the circle, which is the distance between the center C and one of the endpoints, say A(6, 5).

The distance formula between two points A(x_1, y_1) and B(x_2, y_2) is given by:
Distance formula: \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Substitute the coordinates of C(2, 8) and A(6, 5) into the distance formula:
r = \sqrt{(6-2)^2 + (5-8)^2}
r = \sqrt{4^2 + (-3)^2}
r = \sqrt{16 + 9}
r = \sqrt{25}
r = 5

So, the radius of the circle is 5.

Therefore, the equation of the circle is:
(x-2)^2 + (y-8)^2 = 25

\boxed{(x-2)^2 + (y-8)^2 = 25} is the equation of the circle.