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To find the equation of the circle whose diameter has endpoints \((1, -6)\) and \((-3, -4)\), we will follow these steps:

1. **Find the center** of the circle, which is the midpoint of the diameter.
   The midpoint formula is:
   \[
   \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
   \]
   Substituting the values:
   \[
   \text{Center} = \left( \frac{1 + (-3)}{2}, \frac{-6 + (-4)}{2} \right) = \left( \frac{-2}{2}, \frac{-10}{2} \right) = (-1, -5)
   \]
   So, the center of the circle is \((-1, -5)\).

2. **Find the radius** of the circle. The radius is half the length of the diameter. We can calculate the length of the diameter using the distance formula:
   \[
   \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]
   Substituting the values:
   \[
   \text{Distance} = \sqrt{(-3 - 1)^2 + (-4 - (-6))^2} = \sqrt{(-4)^2 + (2)^2} = \sqrt{16 + 4} = \sqrt{20}
   \]
   So, the diameter is \(\sqrt{20}\), and the radius is:
   \[
   \text{Radius} = \frac{\sqrt{20}}{2} = \frac{2\sqrt{5}}{2} = \sqrt{5}
   \]

3. **Write the equation of the circle** using the standard form:
   \[
   (x - h)^2 + (y - k)^2 = r^2
   \]
   Where \((h, k)\) is the center and \(r\) is the radius. Substituting the center \((-1, -5)\) and radius \(\sqrt{5}\), we get:
   \[
   (x + 1)^2 + (y + 5)^2 = (\sqrt{5})^2
   \]
   Simplifying:
   \[
   (x + 1)^2 + (y + 5)^2 = 5
   \]

Thus, the equation of the circle is:
\[
(x + 1)^2 + (y + 5)^2 = 5
\]

Would you like a detailed explanation of any specific part? 

Here are five related questions to expand this concept:
1. How do you calculate the equation of a circle given its radius and center?
2. What is the relationship between the distance formula and the radius of a circle?
3. How would the equation change if the endpoints of the diameter were on different axes?
4. Can you find the equation of a circle if the center and a point on the circle are given?
5. How do transformations affect the equation of a circle?

**Tip:** Always verify whether the given points are correctly placed on the circle by plugging them into the final equation.

Hallar la ecuación del círculo cuyo diámetro tiene extremos (1, -6) y (-3, -4).

Learn how to find the equation of a circle with given diameter endpoints at (1, -6) and (-3, -4). This problem covers key concepts in analytic geometry, including the midpoint formula, distance formula, and the equation of a circle. Suitable for students in Grades 10-12.

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