Given a straight line `x cos 30^@ + y sin 30^@ = 2`. Determine the equation of the other line which is parallel to it and passes through (4, 3). 

Answer

To find the equation of the line parallel to the given line \( x \cos 30^\circ + y \sin 30^\circ = 2 \) that passes through the point \( (4, 3) \), we can follow these steps:

### Step 1: Convert the given line equation to slope-intercept form
We start with the equation:
\[ x \cos 30^\circ + y \sin 30^\circ = 2 \]
Substituting the values of \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) and \( \sin 30^\circ = \frac{1}{2} \):
\[ x \cdot \frac{\sqrt{3}}{2} + y \cdot \frac{1}{2} = 2 \]
Now, we can rearrange this equation to solve for \( y \):
\[ y \cdot \frac{1}{2} = -x \cdot \frac{\sqrt{3}}{2} + 2 \]
Multiplying through by 2 to eliminate the fraction:
\[ y = -\sqrt{3} x + 4 \]

### Step 2: Identify the slope of the given line
From the equation \( y = -\sqrt{3} x + 4 \), we can see that the slope \( m \) of the given line is:
\[ m = -\sqrt{3} \]

### Step 3: Write the equation of the parallel line
Since parallel lines have the same slope, the equation of the line that is parallel to the given line and passes through the point \( (4, 3) \) can be written as:
\[ y = -\sqrt{3} x + c \]
where \( c \) is a constant that we need to determine.

### Step 4: Substitute the point (4, 3) into the equation to find \( c \)
We substitute \( x = 4 \) and \( y = 3 \) into the equation:
\[ 3 = -\sqrt{3} \cdot 4 + c \]
This simplifies to:
\[ 3 = -4\sqrt{3} + c \]
Now, solving for \( c \):
\[ c = 3 + 4\sqrt{3} \]

### Step 5: Write the final equation of the parallel line
Now we can substitute the value of \( c \) back into the equation of the line:
\[ y = -\sqrt{3} x + (3 + 4\sqrt{3}) \]
This can be rearranged to:
\[ y + \sqrt{3} x = 3 + 4\sqrt{3} \]

### Final Answer
The equation of the line parallel to the given line and passing through the point \( (4, 3) \) is:
\[ y + \sqrt{3} x = 3 + 4\sqrt{3} \]