Find the standard form of the equation of the circle with endpoints of a diameter at the points $(5,2)$ and $(-1,6)$.

Step 1 ：Given the endpoints of a diameter at the points (5,2) and (-1,6), we need to find the standard form of the equation of the circle.

Step 2 ：The standard form of the equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\), where \((h,k)\) is the center of the circle and \(r\) is the radius.

Step 3 ：The center of the circle is the midpoint of the diameter, which is given by \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\). Substituting the given points, we find that \(h = 2.0\) and \(k = 4.0\).

Step 4 ：The radius of the circle is half the distance between the two points. The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). Substituting the given points and taking half of the result, we find that \(r = 3.605551275463989\).

Step 5 ：Substituting these values into the standard form of the equation of a circle, we get the equation of our circle as \((x - 2.0)^2 + (y - 4.0)^2 = 12.999999999999998\).

Step 6 ：Simplifying the equation, we get the final answer: The standard form of the equation of the circle with endpoints of a diameter at the points (5,2) and (-1,6) is \(\boxed{(x - 2.0)^2 + (y - 4.0)^2 = 13.0}\).