se of center(10,-12) point on circle(11,-11)

To find the equation of the circle with a center at \((10, -12)\) and passing through the point \((11, -11)\), follow these steps:

### 1. **Identify the center \((h, k)\):**
   - \(h = 10\)
   - \(k = -12\)

### 2. **Find the radius \(r\):**
   The radius \(r\) can be found using the distance formula between the center \((h, k)\) and the given point \((x_1, y_1)\) on the circle:
   \[
   r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2}
   \]
   Substituting the given values:
   \[
   r = \sqrt{(11 - 10)^2 + (-11 + 12)^2} = \sqrt{(1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2}
   \]

### 3. **Write the equation of the circle:**
   The general form of the equation of a circle is:
   \[
   (x - h)^2 + (y - k)^2 = r^2
   \]
   Substituting \(h = 10\), \(k = -12\), and \(r = \sqrt{2}\):
   \[
   (x - 10)^2 + (y + 12)^2 = (\sqrt{2})^2
   \]
   Simplifying:
   \[
   (x - 10)^2 + (y + 12)^2 = 2
   \]

So, the equation of the circle is:
\[
\boxed{(x - 10)^2 + (y + 12)^2 = 2}
\]

Would you like any further details or explanations on this? Here are some related questions you might be interested in:

1. How does the distance formula work in finding the radius?
2. Can the equation of a circle be expanded into a different form?
3. How can you find the coordinates of other points on this circle?
4. How would the equation change if the center of the circle was at the origin?
5. How do you graph a circle with this equation on a coordinate plane?
6. What are the implications of a negative value in the coordinates of the center?
7. How can you use the circle's equation to find the length of the diameter?
8. How would you verify if a given point lies on this circle?

**Tip:** When calculating the radius, always ensure you correctly substitute the coordinates into the distance formula to avoid errors.

Learn how to find the equation of a circle with a center at (10, -12) and passing through the point (11, -11) using the distance formula and circle equation. Understand the process step-by-step with detailed explanations and examples. Suitable for high school students.

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