A linear equation `3x + 4y = 24,` intersects x-axis and y-axis respectively at points A and B. If `P (2, 0) and Q (0, (3)/(2))` respectively lies on the straight line OA and OB, then area of the quadrilateral PABQ is

Options

A. `5/2` sq. unit
B. `(15)/(2)` sq. unit
C. `(35)/(2)` sq. unit
D. `(45)/(2)` sq. unit

Answer

To find the area of the quadrilateral PABQ formed by the points P(2, 0), A, B, and Q(0, 3/2), we will follow these steps:

### Step 1: Find the points A and B where the line intersects the axes.
The given linear equation is:
\[ 3x + 4y = 24 \]

**To find point A (x-intercept):**
Set \( y = 0 \):
\[ 3x + 4(0) = 24 \]
\[ 3x = 24 \]
\[ x = \frac{24}{3} = 8 \]
Thus, point A is \( (8, 0) \).

**To find point B (y-intercept):**
Set \( x = 0 \):
\[ 3(0) + 4y = 24 \]
\[ 4y = 24 \]
\[ y = \frac{24}{4} = 6 \]
Thus, point B is \( (0, 6) \).

### Step 2: Identify the coordinates of points P and Q.
We have:
- Point P = \( (2, 0) \)
- Point Q = \( \left(0, \frac{3}{2}\right) \)

### Step 3: Calculate the area of quadrilateral PABQ.
The area of quadrilateral PABQ can be calculated using the formula for the area of a polygon based on its vertices. The formula for the area of a quadrilateral with vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\), and \((x_4, y_4)\) is given by:

\[ \text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right| \]

Substituting the coordinates of points P, A, B, and Q:
- \( P(2, 0) \)
- \( A(8, 0) \)
- \( B(0, 6) \)
- \( Q(0, \frac{3}{2}) \)

We can assign:
- \( (x_1, y_1) = (2, 0) \)
- \( (x_2, y_2) = (8, 0) \)
- \( (x_3, y_3) = (0, 6) \)
- \( (x_4, y_4) = (0, \frac{3}{2}) \)

Now substituting into the area formula:

\[ \text{Area} = \frac{1}{2} \left| 2 \cdot 0 + 8 \cdot 6 + 0 \cdot \frac{3}{2} + 0 \cdot 0 - (0 \cdot 8 + 0 \cdot 0 + 6 \cdot 0 + \frac{3}{2} \cdot 2) \right| \]

Calculating each term:
\[ = \frac{1}{2} \left| 0 + 48 + 0 + 0 - (0 + 0 + 0 + 3) \right| \]
\[ = \frac{1}{2} \left| 48 - 3 \right| \]
\[ = \frac{1}{2} \left| 45 \right| = \frac{45}{2} \]

### Final Answer:
The area of quadrilateral PABQ is \( \frac{45}{2} \) square units.

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