Center: (10, -12) Point On Circle: (11, -11)

To find the equation of the circle, we need to use the general formula for a circle centered at \((h, k)\) with a radius \(r\):

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Given:
- Center \((h, k) = (10, -12)\)
- A point on the circle \((x_1, y_1) = (11, -11)\)

### Step 1: Calculate the radius \(r\)
The radius \(r\) is the distance between the center and the given point on the circle. We can find this distance using the distance formula:

\[
r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2}
\]

Substituting the given values:

\[
r = \sqrt{(11 - 10)^2 + (-11 - (-12))^2} = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2}
\]

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

### Step 2: Write the equation of the circle
Now that we have the radius \(r\) and the center \((h, k)\), we can substitute these into the general formula:

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

Simplifying:

\[
(x - 10)^2 + (y + 12)^2 = 2
\]

So, the equation of the circle is:

\[
(x - 10)^2 + (y + 12)^2 = 2
\]

Would you like more details on this, or do you have any further questions?

### Related Questions:
1. How do you find the equation of a circle given two points on the circle?
2. What is the equation of a circle in standard form?
3. How can you determine if a point lies inside, outside, or on the circle?
4. How do you convert the general form of a circle equation to standard form?
5. What are the parametric equations for a circle?

**Tip:** When calculating the radius from a center and a point, always ensure that the distance is computed correctly, as even small errors can affect the equation of the circle.

Discover how to find the equation of a circle with a center at (10, -12) and a point on the circle at (11, -11). This problem involves key geometric concepts such as the equation of a circle and the distance formula, making it ideal for students in Grades 9-11.

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