The equation of the line which `p=3, alpha=120^(@)` is

Options

A. `x+sqrt(3)y=6`
B. `x-sqrt(3)y+6=0`
C. `x+sqrt(3)y+6=0`
D. `-x+sqrt(3)y+6=0`

Answer

To find the equation of the line given \( p = 3 \) and \( \alpha = 120^\circ \), we can follow these steps:

### Step 1: Write the equation of the line in normal form
The equation of a line in normal form is given by:
\[ x \cos \alpha + y \sin \alpha = p \]
Substituting the values \( p = 3 \) and \( \alpha = 120^\circ \):
\[ x \cos 120^\circ + y \sin 120^\circ = 3 \]

### Step 2: Calculate \( \cos 120^\circ \) and \( \sin 120^\circ \)
Using trigonometric values:
\[ \cos 120^\circ = -\frac{1}{2} \]
\[ \sin 120^\circ = \frac{\sqrt{3}}{2} \]

### Step 3: Substitute the trigonometric values into the equation
Now substituting these values into the equation:
\[ x \left(-\frac{1}{2}\right) + y \left(\frac{\sqrt{3}}{2}\right) = 3 \]
This simplifies to:
\[ -\frac{x}{2} + \frac{\sqrt{3}}{2}y = 3 \]

### Step 4: Eliminate the fractions
To eliminate the fractions, multiply the entire equation by 2:
\[ - x + \sqrt{3}y = 6 \]

### Step 5: Rearranging the equation
Rearranging the equation gives us:
\[ \sqrt{3}y - x = 6 \]

### Final Answer
Thus, the equation of the line is:
\[ \sqrt{3}y - x = 6 \]