The area (in sq. units) covered by `[x-y]=-3` with the coordinate axes is (where `[.]` is the greatest integer function)

Options

A. 2
B. 4
C. `(5)/(2)`
D. `(11)/(4)`

Answer

To find the area covered by the equation \([x - y] = -3\) with the coordinate axes, we can follow these steps:

### Step 1: Understand the Greatest Integer Function
The expression \([x - y] = -3\) means that \(x - y\) falls within the range of \([-3, -2)\). This means:
\[ -3 \leq x - y < -2 \]

### Step 2: Set Up the Inequalities
From the inequality \(x - y \geq -3\), we can rearrange it to find:
\[ x - y = -3 \implies x = y - 3 \]
From the inequality \(x - y < -2\), we rearrange it to find:
\[ x - y = -2 \implies x = y - 2 \]

### Step 3: Graph the Lines
Now we have two lines:
1. \(x = y - 3\) (or \(y = x + 3\))
2. \(x = y - 2\) (or \(y = x + 2\))

We can plot these lines on a coordinate plane.

### Step 4: Find Intersections with the Axes
To find the area between these two lines and the coordinate axes, we need to find the points where these lines intersect the axes.

- For \(y = x + 3\):
- When \(x = 0\), \(y = 3\) (point (0, 3))
- When \(y = 0\), \(x = -3\) (point (-3, 0))

- For \(y = x + 2\):
- When \(x = 0\), \(y = 2\) (point (0, 2))
- When \(y = 0\), \(x = -2\) (point (-2, 0))

### Step 5: Identify the Shape
The area between these two lines and the coordinate axes forms a trapezium (trapezoid). The vertices of the trapezium are:
- (0, 2)
- (0, 3)
- (-2, 0)
- (-3, 0)

### Step 6: Calculate the Area of the Trapezium
The area \(A\) of a trapezium can be calculated using the formula:
\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]
where \(b_1\) and \(b_2\) are the lengths of the two parallel sides, and \(h\) is the height.

- Length of \(b_1\) (from (0, 2) to (0, 3)): \(3 - 2 = 1\)
- Length of \(b_2\) (from (-2, 0) to (-3, 0)): \(3 - 2 = 1\)
- Height \(h\) (distance between the two lines): \(1\)

Now substituting the values:
\[ A = \frac{1}{2} \times (1 + 1) \times 1 = \frac{1}{2} \times 2 \times 1 = 1 \]

### Final Area Calculation
Thus, the area covered by \([x - y] = -3\) with the coordinate axes is:
\[ \text{Area} = 1 \text{ square unit} \]