To find the **equation of the circle** given the center \((15, 9)\) and a point on the circle \((17, 6)\), we can follow these steps:

### 1. **Identify the center \((h, k)\):**
   - \(h = 15\)
   - \(k = 9\)

### 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:
   \[
   r = \sqrt{(17 - 15)^2 + (6 - 9)^2} = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}
   \]

### 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 \(h = 15\), \(k = 9\), and \(r = \sqrt{13}\):
   \[
   (x - 15)^2 + (y - 9)^2 = (\sqrt{13})^2
   \]
   Simplifying:
   \[
   (x - 15)^2 + (y - 9)^2 = 13
   \]

So, the equation of the circle is:
\[
\boxed{(x - 15)^2 + (y - 9)^2 = 13}
\]

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

1. How do you derive the distance formula used to calculate the radius?
2. What are the general forms of circle equations?
3. How do you convert the circle equation into standard form?
4. How would you find the equation of a circle if the radius was given instead?
5. What are the properties of circles in a coordinate plane?
6. How can you determine if a point lies inside, on, or outside a circle?
7. How would you graph this circle on a coordinate plane?
8. How do transformations affect the equation of a circle?

**Tip:** Remember, the radius is always the distance from the center of the circle to any point on the circle's circumference.

Learn how to find the equation of a circle using the center (15, 9) and a point on the circle (17, 6). Includes step-by-step solution with distance formula and circle equation derivation. Suitable for high school students.

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