A circle in the xy -plane has center ( 5, 7) and radius 2. Which of the following is an equation of the circle?

Options

A. `(x − 5)^2 + (y − 7)^2 = 4`
B. `(x + 5)^2 + (y + 7)^2 = 4`
C. `(x − 5)^2 + (y − 7)^2 = 2`
D. `(x + 5)^2 + (y + 7)^2 = 2`

Explanation

To find the equation of a circle with a given center and radius, we can use the standard form of the equation of a circle, which is:

\[ (x - x_1)^2 + (y - y_1)^2 = r^2 \]

where \((x_1, y_1)\) is the center of the circle and \(r\) is the radius.

### Step-by-step Solution:

1. **Identify the center and radius**:
- The center of the circle is given as \((5, 7)\), which means \(x_1 = 5\) and \(y_1 = 7\).
- The radius of the circle is given as \(2\), so \(r = 2\).

2. **Substitute the values into the equation**:
- We substitute \(x_1\), \(y_1\), and \(r\) into the standard equation of the circle:
\[ (x - 5)^2 + (y - 7)^2 = 2^2 \]

3. **Calculate \(r^2\)**:
- Calculate \(2^2\):
\[ 2^2 = 4 \]

4. **Write the final equation**:
- Now, we can write the equation of the circle:
\[ (x - 5)^2 + (y - 7)^2 = 4 \]

### Final Answer:
The equation of the circle is:
\[ (x - 5)^2 + (y - 7)^2 = 4 \]