A circle has the equation 3 left parenthesis x minus 4 right parenthesis squared plus 3 y squared equals 3.                  Find the center​ (h,k) and radius r of the circle.

We are given the equation of a circle:

\[
3(x - 4)^2 + 3y^2 = 3
\]

### Step 1: Simplify the equation
First, divide the entire equation by 3 to make it more recognizable as the standard form of a circle equation.

\[
(x - 4)^2 + y^2 = 1
\]

### Step 2: Identify the center \((h, k)\) and radius \(r\)
The standard equation of a circle is:

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

By comparing this with our simplified equation:

\[
(x - 4)^2 + y^2 = 1
\]

- The center \((h, k)\) is \((4, 0)\).
- The radius \(r\) is the square root of 1, which is \(r = 1\).

### Final answer:
- **Center**: \((h, k) = (4, 0)\)
- **Radius**: \(r = 1\)

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

Here are 5 related questions for further practice:
1. What is the equation of a circle with a center at \((2, -3)\) and a radius of 5?
2. How would the graph of the circle change if the radius doubled?
3. What happens to the equation if the center of the circle moves to the origin?
4. How do you determine the diameter of a circle from its equation?
5. How do you find the points where the circle intersects the x-axis?

**Tip:** Always simplify equations by removing common factors to better identify important elements like the center and radius.

A circle has the equation 3(x - 4)^2 + 3y^2 = 3. Find the center (h,k) and radius r of the circle.

Learn how to find the center (h,k) and radius r of a circle from the equation 3(x - 4)^2 + 3y^2 = 3. This problem covers the standard equation of a circle and how to simplify it to determine key properties. Suitable for students in Grades 9-11.

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