A line making an angle 30° with positive x-axis at (4, 0). Now it is rotated by an angle 15° in clockwise direction. The equation of line is

Options

A. `x + y - 4 = 0`
B. `x - y - 4 = 0`
C. `(sqrt 3 - 2) x + y + 8 - 4 sqrt 3 = 0`
D. `(2 - sqrt 3) x - y - 8 + 4 sqrt 3 = 0`

Answer

To find the equation of the line that makes an angle of 30° with the positive x-axis at the point (4, 0) and is then rotated 15° clockwise, we can follow these steps:

### Step 1: Determine the new angle of the line
The original angle with the x-axis is 30°. After rotating it 15° clockwise, the new angle θ with the x-axis will be:
\[ \theta = 30° - 15° = 15° \]

### Step 2: Calculate the slope of the line
The slope \( m \) of the line can be calculated using the tangent of the angle:
\[ m = \tan(\theta) = \tan(15°) \]
Using the tangent subtraction formula, we can express \( \tan(15°) \) as:
\[ \tan(15°) = \tan(45° - 30°) = \frac{\tan(45°) - \tan(30°)}{1 + \tan(45°)\tan(30°)} \]
Where \( \tan(45°) = 1 \) and \( \tan(30°) = \frac{1}{\sqrt{3}} \):
\[ \tan(15°) = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \]

### Step 3: Write the equation of the line
We can use the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
Where \( (x_1, y_1) = (4, 0) \):
\[ y - 0 = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}(x - 4) \]
This simplifies to:
\[ y = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}(x - 4) \]

### Step 4: Rearranging the equation
To express this in a more standard form, we can multiply both sides by \( \sqrt{3} + 1 \):
\[ y(\sqrt{3} + 1) = (\sqrt{3} - 1)(x - 4) \]
Expanding both sides gives:
\[ y\sqrt{3} + y = (\sqrt{3} - 1)x - 4(\sqrt{3} - 1) \]
Rearranging leads to:
\[ (\sqrt{3} - 1)x - y\sqrt{3} - y + 4(\sqrt{3} - 1) = 0 \]

### Step 5: Final form of the equation
This can be rearranged to:
\[ (\sqrt{3} - 1)x - y\sqrt{3} - y + 4\sqrt{3} - 4 = 0 \]

### Summary of the Equation
The final equation of the line after the rotation is:
\[ (\sqrt{3} - 1)x - \sqrt{3}y + 4\sqrt{3} - 4 = 0 \]