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To find the equation of the circle whose diameter has endpoints at \((-1, 6)\) and \((5, -2)\), follow these steps:

1. **Find the center of the circle**:
   The center is the midpoint of the diameter. To find the midpoint, use the midpoint formula:

   \[
   \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
   \]
   Plugging in the coordinates \((-1, 6)\) and \((5, -2)\):

   \[
   \text{Midpoint} = \left( \frac{-1 + 5}{2}, \frac{6 + (-2)}{2} \right) = \left( \frac{4}{2}, \frac{4}{2} \right) = (2, 2)
   \]

   So, the center of the circle is at \((2, 2)\).

2. **Find the radius**:
   The radius is half the length of the diameter. To find the length of the diameter, use the distance formula:

   \[
   \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]
   Plugging in the coordinates:

   \[
   \text{Distance} = \sqrt{(5 - (-1))^2 + (-2 - 6)^2} = \sqrt{(5 + 1)^2 + (-2 - 6)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10
   \]

   The diameter is 10, so the radius is:

   \[
   r = \frac{10}{2} = 5
   \]

3. **Write the equation of the circle**:
   The general form of the equation of a circle is:

   \[
   (x - h)^2 + (y - k)^2 = r^2
   \]
   where \((h, k)\) is the center and \(r\) is the radius. Substituting the center \((2, 2)\) and radius \(5\):

   \[
   (x - 2)^2 + (y - 2)^2 = 5^2
   \]
   Simplifying:

   \[
   (x - 2)^2 + (y - 2)^2 = 25
   \]

Thus, the equation of the circle is:

\[
(x - 2)^2 + (y - 2)^2 = 25
\]

Would you like more details or further clarification on any part of this?

Here are some related questions to deepen your understanding:
1. How would you find the equation of a circle if only the center and one point on the circle are given?
2. How does the general equation of a circle change if the circle is centered at the origin?
3. Can you derive the formula for the area of a circle from its equation?
4. How would you modify the equation of a circle if it is shifted horizontally or vertically?
5. What are the different forms of representing the equation of a circle?

**Tip**: Always verify if the center and radius found from the given information correctly satisfy the equation of the circle.

Find an equation of the circle whose diameter has endpoints (-1, 6) and (5, -2).

Learn how to find the equation of a circle with endpoints of its diameter at (-1, 6) and (5, -2). This problem covers key geometry concepts, including the midpoint and distance formulas, and demonstrates how to apply the equation of a circle.

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