Let P(x,y) be a moving point in the x-y plane such that [x].[y] = 2, where [.] denotes the greatest inte- ger function, then area of the region containing the points P(x,y) is equal to :

Options

A. 1 sq. units
B. 2 sq. units
C. 4 sq. units
D. None of these

Answer

To solve the problem, we need to find the area of the region defined by the condition \([x][y] = 2\), where \([.]\) denotes the greatest integer function.

### Step-by-Step Solution:

1. **Understanding the Greatest Integer Function**:
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). Therefore, \([x][y] = 2\) implies that the product of the greatest integers of \(x\) and \(y\) must equal 2.

2. **Identifying Integer Pairs**:
We need to find pairs of integers \((m, n)\) such that \(m \cdot n = 2\). The possible pairs are:
- \( (1, 2) \)
- \( (2, 1) \)
- \( (-1, -2) \)
- \( (-2, -1) \)

3. **Finding Ranges for Each Pair**:
For each pair, we can determine the ranges of \(x\) and \(y\):
- For \( (1, 2) \):
- \([x] = 1 \implies 1 \leq x < 2\)
- \([y] = 2 \implies 2 \leq y < 3\)
- This gives the rectangle defined by \(1 \leq x < 2\) and \(2 \leq y < 3\).

- For \( (2, 1) \):
- \([x] = 2 \implies 2 \leq x < 3\)
- \([y] = 1 \implies 1 \leq y < 2\)
- This gives the rectangle defined by \(2 \leq x < 3\) and \(1 \leq y < 2\).

- For \( (-1, -2) \):
- \([x] = -1 \implies -2 < x \leq -1\)
- \([y] = -2 \implies -3 < y \leq -2\)
- This gives the rectangle defined by \(-2 < x \leq -1\) and \(-3 < y \leq -2\).

- For \( (-2, -1) \):
- \([x] = -2 \implies -3 < x \leq -2\)
- \([y] = -1 \implies -2 < y \leq -1\)
- This gives the rectangle defined by \(-3 < x \leq -2\) and \(-2 < y \leq -1\).

4. **Calculating the Area of Each Rectangle**:
Each rectangle has a width and height of 1 unit:
- Area of rectangle for \( (1, 2) \): \(1 \times 1 = 1\)
- Area of rectangle for \( (2, 1) \): \(1 \times 1 = 1\)
- Area of rectangle for \( (-1, -2) \): \(1 \times 1 = 1\)
- Area of rectangle for \( (-2, -1) \): \(1 \times 1 = 1\)

5. **Total Area**:
The total area is the sum of the areas of all rectangles:
\[ \text{Total Area} = 1 + 1 + 1 + 1 = 4 \text{ square units} \]

### Final Answer:
The area of the region containing the points \(P(x, y)\) is \(4\) square units.

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