A dog wants to catch a cat. The dog follows the path whose equation is `y-x=0` while the cat follows the path whose equation is `x^(2)+y^(2)=8`. The coordinates of possible points of catching the cat are:

Options

A. `(2, -2)`
B. `(2, 2)`
C. `(-2, 2)`
D. `(-2, -2)`

Explanation

To solve the problem of finding the coordinates of possible points where the dog can catch the cat, we need to analyze the paths they are following mathematically.

### Step-by-Step Solution:

1. **Identify the Paths**:
- The dog's path is given by the equation \( y - x = 0 \), which simplifies to \( y = x \). This represents a straight line passing through the origin at a 45-degree angle.
- The cat's path is given by the equation \( x^2 + y^2 = 8 \). This represents a circle centered at the origin with a radius of \( \sqrt{8} = 2\sqrt{2} \).

2. **Substitute the Dog's Path into the Cat's Path**:
- Since the dog moves along the line \( y = x \), we can substitute \( y \) with \( x \) in the cat's equation:
\[ x^2 + y^2 = 8 \implies x^2 + x^2 = 8 \]
- This simplifies to:
\[ 2x^2 = 8 \]

3. **Solve for \( x \)**:
- Divide both sides by 2:
\[ x^2 = 4 \]
- Taking the square root of both sides gives:
\[ x = \pm 2 \]

4. **Find Corresponding \( y \) Values**:
- Since \( y = x \), we have:
- If \( x = 2 \), then \( y = 2 \).
- If \( x = -2 \), then \( y = -2 \).

5. **Write the Coordinates of Possible Points**:
- The possible points where the dog can catch the cat are:
- Point A: \( (2, 2) \)
- Point B: \( (-2, -2) \)

### Final Answer:
The coordinates of possible points of catching the cat are \( (2, 2) \) and \( (-2, -2) \).