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 where the dog can catch the cat, we need to analyze the equations given for their paths.

### Step-by-Step Solution:

1. **Identify the equations of the paths:**
- The dog's path is given by the equation:
\[ y - x = 0 \quad \text{(Equation 1)} \]
- The cat's path is given by the equation:
\[ x^2 + y^2 = 8 \quad \text{(Equation 2)} \]

2. **Express one variable in terms of the other:**
- From Equation 1, we can express \(y\) in terms of \(x\):
\[ y = x \]

3. **Substitute \(y\) into the cat's equation:**
- Substitute \(y = x\) into Equation 2:
\[ x^2 + (x)^2 = 8 \]
- This simplifies to:
\[ 2x^2 = 8 \]

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

5. **Find the corresponding \(y\) values:**
- Since \(y = x\), we have:
- If \(x = 2\), then \(y = 2\).
- If \(x = -2\), then \(y = -2\).

6. **List the coordinates of possible catching points:**
- The possible coordinates where the dog can catch the cat are:
\[ (2, 2) \quad \text{and} \quad (-2, -2) \]

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

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