Show on a diagram the position of the straight line `x"cos"30^(@)+y"sin"30^(@)=2` in relation to the co-ordinate axes, indicating clearly which angle is `30^(@)` and which length is 2 units. Find
(i) the equation of the straight line parallel to that given line and passing through the point (4, 3) and
(ii) the distance between the two parallel straight lines.

Answer

To solve the problem step by step, we will first analyze the given equation of the line, then draw the diagram, find the equation of the parallel line, and finally calculate the distance between the two parallel lines.

### Step 1: Analyze the Given Line Equation
The given line equation is:
\[ x \cos 30^\circ + y \sin 30^\circ = 2 \]

Using the values of \(\cos 30^\circ\) and \(\sin 30^\circ\):
\[ \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2} \]
Substituting these values into the equation:
\[ x \cdot \frac{\sqrt{3}}{2} + y \cdot \frac{1}{2} = 2 \]
Multiplying through by 2 to eliminate the fractions:
\[ \sqrt{3} x + y = 4 \]

### Step 2: Find Intercepts for the Diagram
To plot the line, we need to find the x-intercept and y-intercept.

1. **Finding the y-intercept** (set \(x = 0\)):
\[ \sqrt{3}(0) + y = 4 \implies y = 4 \]
So, the y-intercept is \( (0, 4) \).

2. **Finding the x-intercept** (set \(y = 0\)):
\[ \sqrt{3} x + 0 = 4 \implies x = \frac{4}{\sqrt{3}} \]
So, the x-intercept is \( \left( \frac{4}{\sqrt{3}}, 0 \right) \).

### Step 3: Draw the Diagram
On the coordinate axes:
- Mark the point \( (0, 4) \) on the y-axis.
- Mark the point \( \left( \frac{4}{\sqrt{3}}, 0 \right) \) on the x-axis.
- Draw the line connecting these two points.
- Indicate the angle \( 30^\circ \) with respect to the x-axis and label the length of 2 units.

### Step 4: Find the Equation of the Parallel Line
The slope of the given line is the coefficient of \(x\) when rearranged in slope-intercept form:
\[ y = -\sqrt{3} x + 4 \]
Thus, the slope \(m = -\sqrt{3}\).

To find the equation of the line parallel to this and passing through the point \( (4, 3) \):
Using the point-slope form:
\[ y - y_1 = m(x - x_1) \]
Substituting \(m = -\sqrt{3}\), \(x_1 = 4\), and \(y_1 = 3\):
\[ y - 3 = -\sqrt{3}(x - 4) \]
Expanding this:
\[ y - 3 = -\sqrt{3}x + 4\sqrt{3} \]
Rearranging gives:
\[ \sqrt{3}x + y - (4\sqrt{3} + 3) = 0 \]

### Step 5: Calculate the Distance Between the Two Parallel Lines
The distance \(d\) between two parallel lines given by \(y = mx + c_1\) and \(y = mx + c_2\) is given by:
\[ d = \frac{|c_2 - c_1|}{\sqrt{1 + m^2}} \]
Here, \(c_1 = 4\) and \(c_2 = 4\sqrt{3} + 3\). Thus:
\[ d = \frac{|(4\sqrt{3} + 3) - 4|}{\sqrt{1 + (-\sqrt{3})^2}} = \frac{|4\sqrt{3} - 1|}{\sqrt{1 + 3}} = \frac{|4\sqrt{3} - 1|}{2} \]

### Final Answers
1. The equation of the parallel line is:
\[ \sqrt{3}x + y - (4\sqrt{3} + 3) = 0 \]
2. The distance between the two parallel lines is:
\[ d = \frac{|4\sqrt{3} - 1|}{2} \]