Area of triangle formed by straight line `4x - 3y + 4 =0, 4x + 3y - 20 = 0` and x-axis is

Options

A. 3 sq. unit
B. 6 sq. unit
C. 12 sq. unit
D. 24 sq. unit

Answer

To find the area of the triangle formed by the lines \(4x - 3y + 4 = 0\), \(4x + 3y - 20 = 0\), and the x-axis, we can follow these steps:

### Step 1: Find the x-intercepts of both lines.
To find the x-intercept, we set \(y = 0\) in both equations.

1. For the first line \(4x - 3y + 4 = 0\):
\[ 4x + 4 = 0 \implies 4x = -4 \implies x = -1 \]
So, the x-intercept is \((-1, 0)\).

2. For the second line \(4x + 3y - 20 = 0\):
\[ 4x - 20 = 0 \implies 4x = 20 \implies x = 5 \]
So, the x-intercept is \((5, 0)\).

### Step 2: Find the y-intercepts of both lines.
To find the y-intercept, we set \(x = 0\) in both equations.

1. For the first line \(4x - 3y + 4 = 0\):
\[ -3y + 4 = 0 \implies -3y = -4 \implies y = \frac{4}{3} \]
So, the y-intercept is \((0, \frac{4}{3})\).

2. For the second line \(4x + 3y - 20 = 0\):
\[ 3y - 20 = 0 \implies 3y = 20 \implies y = \frac{20}{3} \]
So, the y-intercept is \((0, \frac{20}{3})\).

### Step 3: Identify the vertices of the triangle.
The vertices of the triangle formed by the two lines and the x-axis are:
- \((-1, 0)\) from the first line.
- \((5, 0)\) from the second line.
- \((0, \frac{4}{3})\) from the first line.

### Step 4: Calculate the area of the triangle.
The area \(A\) of a triangle formed by vertices at \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Substituting the vertices \((-1, 0)\), \((5, 0)\), and \((0, \frac{4}{3})\):
\[ A = \frac{1}{2} \left| -1(0 - \frac{4}{3}) + 5(\frac{4}{3} - 0) + 0(0 - 0) \right| \]
\[ = \frac{1}{2} \left| -1 \cdot -\frac{4}{3} + 5 \cdot \frac{4}{3} \right| \]
\[ = \frac{1}{2} \left| \frac{4}{3} + \frac{20}{3} \right| \]
\[ = \frac{1}{2} \left| \frac{24}{3} \right| = \frac{1}{2} \cdot 8 = 4 \]

### Step 5: Calculate the base and height.
The base of the triangle is the distance between the x-intercepts:
\[ \text{Base} = 5 - (-1) = 6 \]
The height is the y-coordinate of the intersection point with the y-axis:
\[ \text{Height} = \frac{4}{3} \]

### Step 6: Calculate the area using base and height.
Using the area formula for a triangle:
\[ A = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 6 \times \frac{4}{3} = \frac{1}{2} \times 8 = 4 \]

### Final Answer:
The area of the triangle formed by the lines and the x-axis is \(12\) square units.

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