write down the equation of the line for which p=3, alpha=120 degree

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:
\[ x \cos \alpha + y \sin \alpha = p \]

### Step 2: Substitute the values of \( p \) and \( \alpha \)
Here, we know \( p = 3 \) and \( \alpha = 120^\circ \). We need to find \( \cos 120^\circ \) and \( \sin 120^\circ \).

### Step 3: 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 4: Substitute these values into the equation
Now we substitute these values into the normal form equation:
\[ x \left(-\frac{1}{2}\right) + y \left(\frac{\sqrt{3}}{2}\right) = 3 \]

### Step 5: Simplify the equation
This simplifies to:
\[ -\frac{1}{2}x + \frac{\sqrt{3}}{2}y = 3 \]

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

### Step 7: Rearrange the equation
Rearranging gives us:
\[ -x + \sqrt{3}y - 6 = 0 \]

### Step 8: Multiply by -1 (optional)
To write it in a more standard form, we can multiply the entire equation by -1:
\[ x - \sqrt{3}y + 6 = 0 \]

### Final Answer
Thus, the required equation of the line is:
\[ x - \sqrt{3}y + 6 = 0 \]

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