A circle is tangent to the lines with the equations x = 5 and y = 7. Which of the following could be the coordinates of the center of the circle ?

Options

A. (3, 7)
B. (8, 4)
C. (10, 8)
D. (10, 14)

Explanation

To find the coordinates of the center of a circle that is tangent to the lines \( x = 5 \) and \( y = 7 \), we need to determine the distances from the center of the circle to these lines. The distances must be equal because the circle is tangent to both lines.

### Step-by-Step Solution:

1. **Understanding the Tangent Lines**:
- The line \( x = 5 \) is a vertical line.
- The line \( y = 7 \) is a horizontal line.
- The center of the circle will be at some point \( (h, k) \).

2. **Distance from the Center to the Line \( x = 5 \)**:
- The distance from a point \( (h, k) \) to the line \( x = 5 \) is given by:
\[ d_1 = |h - 5| \]

3. **Distance from the Center to the Line \( y = 7 \)**:
- The distance from a point \( (h, k) \) to the line \( y = 7 \) is given by:
\[ d_2 = |k - 7| \]

4. **Setting the Distances Equal**:
- Since the circle is tangent to both lines, we set the distances equal:
\[ |h - 5| = |k - 7| \]

5. **Analyzing Possible Options**:
- We will check the given options to see which one satisfies the equation \( |h - 5| = |k - 7| \).

6. **Checking the First Option \( (3, 7) \)**:
- For \( (3, 7) \):
\[ d_1 = |3 - 5| = 2 \]
\[ d_2 = |7 - 7| = 0 \]
- Since \( d_1 \neq d_2 \), this option is not valid.

7. **Checking the Second Option \( (8, 4) \)**:
- For \( (8, 4) \):
\[ d_1 = |8 - 5| = 3 \]
\[ d_2 = |4 - 7| = 3 \]
- Since \( d_1 = d_2 \), this option is valid.

8. **Checking Other Options** (if any):
- If there are other options provided, repeat the distance calculations for each until you find the one that satisfies \( |h - 5| = |k - 7| \).

### Conclusion:
The coordinates of the center of the circle that is tangent to the lines \( x = 5 \) and \( y = 7 \) could be \( (8, 4) \).