a Find equation of a straight line on which length of perpendicular from the origin is four units and the line makes an angle of `120^0` with the positive direction of x-axis.

Explanation

To find the equation of a straight line that has a perpendicular distance of 4 units from the origin and makes an angle of \(120^\circ\) with the positive direction of the x-axis, we can use the formula for the equation of a line in terms of its angle and perpendicular distance.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Perpendicular distance from the origin, \( r = 4 \)
- Angle with the positive x-axis, \( \theta = 120^\circ \)

2. **Convert the Angle to Radians (if necessary):**
- However, we can directly use the trigonometric functions for degrees in this case.

3. **Use the Formula for the Equation of the Line:**
The general form of the equation of a line in terms of the angle \( \theta \) and the perpendicular distance \( r \) is given by:
\[ x \cos(\theta) + y \sin(\theta) = r \]
Here, \( \theta = 120^\circ \).

4. **Calculate \( \cos(120^\circ) \) and \( \sin(120^\circ) \):**
- \( \cos(120^\circ) = -\frac{1}{2} \)
- \( \sin(120^\circ) = \frac{\sqrt{3}}{2} \)

5. **Substitute the Values into the Equation:**
\[ x \left(-\frac{1}{2}\right) + y \left(\frac{\sqrt{3}}{2}\right) = 4 \]

6. **Multiply through by 2 to eliminate the fractions:**
\[ -x + y\sqrt{3} = 8 \]

7. **Rearranging the Equation:**
\[ y\sqrt{3} = x + 8 \]
or
\[ y = \frac{1}{\sqrt{3}}x + \frac{8}{\sqrt{3}} \]

8. **Final Form of the Equation:**
The equation of the line can also be expressed in standard form:
\[ x - \sqrt{3}y + 8 = 0 \]

### Final Answer:
The equation of the line is:
\[ x - \sqrt{3}y + 8 = 0 \]