Let P = (-1, 0), Q = (0, 0) and R = (3, `3sqrt3`) be three points. The equation of the bisector of the angle PQR

Options

A. `(sqrt(3)//2)x+y= 0`
B. `x+sqrt(3)y= 0`
C. `sqrt(3)x+y= 0`
D. `x+(sqrt(3)//2)y= 0`

Explanation

To find the equation of the angle bisector of the angle PQR formed by the points P = (-1, 0), Q = (0, 0), and R = (3, 3√3), we will follow these steps:

### Step 1: Identify the Points
We have three points:
- P = (-1, 0)
- Q = (0, 0)
- R = (3, 3√3)

### Step 2: Calculate the Slopes of Lines PQ and QR
1. **Slope of line QR**:
- Using the formula for slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \):
\[ m_{QR} = \frac{3\sqrt{3} - 0}{3 - 0} = \frac{3\sqrt{3}}{3} = \sqrt{3} \]

2. **Slope of line PQ**:
- Using the same formula:
\[ m_{PQ} = \frac{0 - 0}{0 - (-1)} = \frac{0}{1} = 0 \]

### Step 3: Determine the Angles
1. **Angle at Q (PQR)**:
- The slope of line QR is \( \sqrt{3} \), which corresponds to an angle \( \theta \) where:
\[ \tan(\theta) = \sqrt{3} \implies \theta = \frac{\pi}{3} \text{ (or 60 degrees)} \]
- The angle PQR is the angle between the line PQ (which is horizontal) and line QR. Since line PQ is horizontal, the angle PQR is:
\[ \text{Angle PQR} = 180^\circ - 60^\circ = 120^\circ = \frac{2\pi}{3} \]

### Step 4: Find the Angle Bisector
1. **Angle Bisector**:
- The angle bisector divides the angle PQR into two equal parts:
\[ \text{Angle of bisector} = \frac{1}{2} \times \frac{2\pi}{3} = \frac{\pi}{3} \]

### Step 5: Determine the Slope of the Bisector
1. **Slope of the Bisector**:
- The slope of the angle bisector can be found using:
\[ m = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

### Step 6: Write the Equation of the Bisector
1. **Equation of the line**:
- Since the angle bisector passes through point Q (0, 0), we can use the point-slope form of the line:
\[ y - y_1 = m(x - x_1) \]
- Substituting \( (x_1, y_1) = (0, 0) \) and \( m = \sqrt{3} \):
\[ y - 0 = \sqrt{3}(x - 0) \implies y = \sqrt{3}x \]

2. **Rearranging to standard form**:
- Rearranging gives:
\[ -\sqrt{3}x + y = 0 \implies \sqrt{3}x - y = 0 \]

### Final Answer
The equation of the bisector of the angle PQR is:
\[ \sqrt{3}x - y = 0 \]