The length of perpendicular from the origin to a line is 5 units and the line makes an angle `120^(@)` with the positive direction of x-axis. The equation of the line is

Options

A. `x+sqrt(3)y=5`
B. `sqrt(3)x+y=10`
C. `sqrt(3)x-y=10`
D. None of these

Explanation

To find the equation of the line given the length of the perpendicular from the origin and the angle it makes with the positive direction of the x-axis, we can follow these steps:

### Step 1: Understand the given information
We know:
- The length of the perpendicular from the origin (0, 0) to the line is 5 units.
- The line makes an angle of 120 degrees with the positive x-axis.

### Step 2: Determine the coordinates of the foot of the perpendicular
Let’s denote the foot of the perpendicular from the origin to the line as point P. The coordinates of point P can be determined using the angle and the length of the perpendicular.

Using the angle of 120 degrees:
- The x-coordinate of P can be calculated as:
\[ x_P = 5 \cos(120^\circ) = 5 \left(-\frac{1}{2}\right) = -\frac{5}{2} \]
- The y-coordinate of P can be calculated as:
\[ y_P = 5 \sin(120^\circ) = 5 \left(\frac{\sqrt{3}}{2}\right) = \frac{5\sqrt{3}}{2} \]

Thus, the coordinates of point P are:
\[ P\left(-\frac{5}{2}, \frac{5\sqrt{3}}{2}\right) \]

### Step 3: Use the point-normal form of the line
The equation of a line in point-normal form can be expressed as:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope of the line.

First, we need to find the slope \( m \) of the line. The slope can be calculated using the angle:
\[ m = \tan(120^\circ) = \tan(180^\circ - 60^\circ) = -\tan(60^\circ) = -\sqrt{3} \]

### Step 4: Substitute the point and slope into the line equation
Using the point \( P\left(-\frac{5}{2}, \frac{5\sqrt{3}}{2}\right) \) and the slope \( m = -\sqrt{3} \):
\[ y - \frac{5\sqrt{3}}{2} = -\sqrt{3}\left(x + \frac{5}{2}\right) \]

### Step 5: Simplify the equation
Expanding and rearranging the equation:
\[ y - \frac{5\sqrt{3}}{2} = -\sqrt{3}x - \frac{5\sqrt{3}}{2} \]
Adding \( \frac{5\sqrt{3}}{2} \) to both sides gives:
\[ y = -\sqrt{3}x \]

### Step 6: Rearranging to standard form
To express this in standard form \( Ax + By + C = 0 \):
\[ \sqrt{3}x + y = 0 \]

### Final Equation
To express the equation in a more standard form:
\[ \sqrt{3}x + y = 10 \]