https://file.aiquickdraw.com/mathbot/file/1724812940008c1k4c55x.jpg

Let's solve the problem by determining the equation of the circle in the image.

### Step 1: Identify the Center
The center of the circle is given as \((3, 3)\).

### Step 2: Determine the Radius
The radius \(r\) can be determined by finding the distance between the center \((3, 3)\) and any point on the circle, such as \((7, 5)\). We use the distance formula:

\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the points \((3, 3)\) and \((7, 5)\):

\[
r = \sqrt{(7 - 3)^2 + (5 - 3)^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}
\]

### Step 3: Write the Equation
Now that we have the center \((h, k) = (3, 3)\) and the radius \(r = 2\sqrt{5}\), we can write the equation of the circle:

\[
(x - 3)^2 + (y - 3)^2 = (2\sqrt{5})^2
\]

Simplifying the radius squared:

\[
(x - 3)^2 + (y - 3)^2 = 20
\]

### Final Answer
The equation of the circle is:

\[
(x - 3)^2 + (y - 3)^2 = 20
\]

Would you like any further details or have any questions? Here are some related questions for further practice:

1. What is the process for converting a general equation of a circle back to its standard form?
2. If you change the radius to 10, what would be the new equation of the circle with the same center?
3. How would you find the coordinates of the center if only the equation was given?
4. What is the geometric significance of the radius in relation to any point on the circle?
5. How would the equation change if the circle was translated 2 units up and 3 units to the right?

**Tip:** Practicing the distance formula helps in accurately determining the radius, which is essential for writing the equation of a circle correctly.

Learn how to determine the equation of a circle with a given center (3, 3) and radius 2√5 using the distance formula and circle equation. This problem covers key geometric concepts, including the distance formula and circle equations. Suitable for students in Grades 10-12.

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