To find the equation of a circle given the endpoints of its diameter, we will follow these steps:

Given:

Endpoints of the diameter are $(1, -5)$ and $(-5, -3)$.

Step 1: Find the center of the circle

The center of the circle is the midpoint of the diameter.
The midpoint formula is: $\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ Using $(1, -5)$ and $(-5, -3)$: $\text{Center} = \left( \frac{1 + (-5)}{2}, \frac{-5 + (-3)}{2} \right) = \left( \frac{-4}{2}, \frac{-8}{2} \right) = (-2, -4)$

So, the center of the circle is at $(-2, -4)$.

Step 2: Find the radius

The radius is half the length of the diameter.
We first calculate the length of the diameter using the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substitute the points $(1, -5)$ and $(-5, -3)$: $d = \sqrt{(-5 - 1)^2 + (-3 + 5)^2} = \sqrt{(-6)^2 + (2)^2} = \sqrt{36 + 4} = \sqrt{40}$ Since the radius $r$ is half the diameter: $r = \frac{\sqrt{40}}{2} = \frac{2\sqrt{10}}{2} = \sqrt{10}$

Step 3: Write the equation of the circle

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

Here, the center is $(-2, -4)$ and the radius $r = \sqrt{10}$.
So: $(x - (-2))^2 + (y - (-4))^2 = (\sqrt{10})^2$ Simplify: $(x + 2)^2 + (y + 4)^2 = 10$

Final Answer:

The equation of the circle is: $(x + 2)^2 + (y + 4)^2 = 10$

Would you like any further explanation?

Here are 5 related questions you might explore:

1. How do you find the equation of a circle if you are given the radius and center directly?
2. What is the general form of a circle's equation, and how do you convert it to standard form?
3. How do you derive the equation of a circle that passes through three points?
4. What changes in the circle’s equation if the radius doubles?
5. How do you calculate the area and circumference of the circle given its equation?

Tip: When solving problems involving diameters or circles, always visualize them on a coordinate plane to avoid mistakes with signs.