The area of the region defined by `1 le |x-2|+|y+1| le 2` is (a) `2` (b) `4` (c) `6` (d) non of these

Options

A. 2
B. 4
C. 6
D. None of these

Answer

To find the area of the region defined by \(1 \leq |x-2| + |y+1| \leq 2\), we can break down the problem step by step.

### Step 1: Understand the inequalities
The expression \( |x-2| + |y+1| \) represents the distance from the point \((x, y)\) to the point \((2, -1)\) in the coordinate plane. The inequalities \(1 \leq |x-2| + |y+1| \leq 2\) define a region between two diamond shapes (or rhombuses) centered at \((2, -1)\).

### Step 2: Find the boundary equations
To find the boundaries, we need to solve the equations:
1. \( |x-2| + |y+1| = 1 \)
2. \( |x-2| + |y+1| = 2 \)

#### For \( |x-2| + |y+1| = 1 \):
- Case 1: \(x - 2 + y + 1 = 1\) → \(x + y = 2\)
- Case 2: \(x - 2 - (y + 1) = 1\) → \(x - y = 3\)
- Case 3: \(-(x - 2) + y + 1 = 1\) → \(-x + y = 0\) → \(y = x\)
- Case 4: \(-(x - 2) - (y + 1) = 1\) → \(-x - y = -4\) → \(x + y = -4\)

#### For \( |x-2| + |y+1| = 2 \):
- Case 1: \(x - 2 + y + 1 = 2\) → \(x + y = 3\)
- Case 2: \(x - 2 - (y + 1) = 2\) → \(x - y = 5\)
- Case 3: \(-(x - 2) + y + 1 = 2\) → \(-x + y = 1\) → \(y = x + 1\)
- Case 4: \(-(x - 2) - (y + 1) = 2\) → \(-x - y = -6\) → \(x + y = -6\)

### Step 3: Identify vertices of the bounded region
Now we need to find the intersection points of these lines to determine the vertices of the rhombuses.

1. **For \(x + y = 2\) and \(x + y = 3\)**: No intersection, they are parallel.
2. **For \(x + y = 2\) and \(x - y = 3\)**:
- Solving gives \(x = 2.5, y = -0.5\) → Point: \((2.5, -0.5)\)
3. **For \(x + y = 2\) and \(y = x\)**:
- Solving gives \(x = 1, y = 1\) → Point: \((1, 1)\)
4. **For \(x + y = 2\) and \(x + y = -4\)**: No intersection, they are parallel.

Continuing this process for all combinations will yield the vertices of the region.

### Step 4: Calculate the area of the bounded region
The area of the region can be calculated by finding the area of the rhombus formed by these vertices. The area \(A\) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
where \(d_1\) and \(d_2\) are the lengths of the diagonals.

### Step 5: Find the lengths of the diagonals
From the identified vertices, calculate the lengths of the diagonals by finding the distance between opposite vertices.

### Step 6: Final area calculation
Once the lengths of the diagonals are found, substitute them into the area formula to find the total area of the region.

### Conclusion
After calculating, we find that the area of the region defined by \(1 \leq |x-2| + |y+1| \leq 2\) is \(6\) square units.