To determine the points at which the predator and prey meet, we need to analyze their respective paths mathematically. ### Step 1: Define the paths of the predator and prey 1. **Predator's Path**: The predator moves in a circle with a radius of \(\sqrt{2}\) centered at the origin (0,0). The equation of a circle is given by: \[ x^2 + y^2 = r^2 \] For our predator, substituting \(r = \sqrt{2}\): \[ x^2 + y^2 = (\sqrt{2})^2 = 2 \] 2. **Prey's Path**: The prey moves along the line defined by the equation \(y = x\). This means for any point on this line, the y-coordinate is equal to the x-coordinate. ### Step 2: Substitute the prey's path into the predator's equation To find the points of intersection, we substitute \(y = x\) into the predator's equation: \[ x^2 + (x)^2 = 2 \] This simplifies to: \[ 2x^2 = 2 \] Dividing both sides by 2 gives: \[ x^2 = 1 \] Taking the square root of both sides, we find: \[ x = 1 \quad \text{or} \quad x = -1 \] ### Step 3: Find corresponding y-coordinates Since \(y = x\), we can find the corresponding y-coordinates: - If \(x = 1\), then \(y = 1\). Thus, one point of intersection is \((1, 1)\). - If \(x = -1\), then \(y = -1\). Thus, another point of intersection is \((-1, -1)\). ### Step 4: Check the range of the prey's movement The problem states that the prey moves along the line \(y = x\) for \(0 \leq x \leq 2\). This means we only consider points where \(x\) is non-negative. Therefore, the point \((-1, -1)\) is not valid since it falls outside the specified range. ### Step 5: Valid points of intersection The only valid point of intersection within the specified range is: \[ (1, 1) \] ### Step 6: Evaluate the options Now, let's evaluate the provided options: - **A. (1,1) only**: This is correct as it is the only valid intersection point. - **B. (1,1) and (1,2)**: Incorrect. While (1,1) is valid, (1,2) does not satisfy the predator's circular path. - **C. (0,0) and (1,1)**: Incorrect. (0,0) is not on the line \(y = x\) for \(0 \leq x \leq 2\). - **D. (√2,√2) only**: Incorrect. While (√2,√2) lies on the circle, it does not lie on the line \(y = x\) for the specified range. ### Summary of the Correct Answer The correct answer is **A. (1,1) only**. ### Revision Summary - The predator moves in a circle defined by \(x^2 + y^2 = 2\). - The prey moves along the line \(y = x\) for \(0 \leq x \leq 2\). - The intersection point found is (1,1), which is valid within the prey's movement range. - Other options either include points outside the valid range or points that do not satisfy both equations.