To find the equation of a circle when you're given the center and a point on the circumference, follow these steps:

1. **Center of the circle**: \((10, -12)\)
2. **Point on the circle**: \((11, -11)\)

### Step 1: Formula of a circle
The general formula for 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 2: Find the radius
The radius \(r\) is the distance between the center \((10, -12)\) and the point \((11, -11)\) on the circumference. Use the distance formula to find this:

\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the values:
\[
r = \sqrt{(11 - 10)^2 + (-11 + 12)^2} = \sqrt{(1)^2 + (1)^2} = \sqrt{2}
\]

So, the radius \(r = \sqrt{2}\).

### Step 3: Write the equation
Now, substitute the center \((h, k) = (10, -12)\) and the radius \(r = \sqrt{2}\) into the circle equation:

\[
(x - 10)^2 + (y + 12)^2 = (\sqrt{2})^2
\]
Simplifying:
\[
(x - 10)^2 + (y + 12)^2 = 2
\]

This is the equation of the circle.

### Final answer:
The equation of the circle is:
\[
(x - 10)^2 + (y + 12)^2 = 2
\]

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the equation of a circle with center \((3, -4)\) and radius 5?
2. How do you find the equation of a circle given two points on the circumference and the center?
3. What is the radius of a circle if the distance between the center and a point on the circle is \(\sqrt{5}\)?
4. Can the center of a circle be outside of the first quadrant if its equation is \((x - 4)^2 + (y - 5)^2 = 16\)?
5. How do you find the equation of a circle if the radius and center are given in decimal form?

**Tip**: Always double-check the distance between the center and any point on the circumference to ensure the correct radius is used!

Learn how to find the equation of a circle with center (10, -12) and a point on the circumference (11, -11). This step-by-step guide uses the circle equation and the distance formula, ideal for students in Grades 8-10.

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