Find the equation of a line which intersects `X`-axis at a distance of `2` units on right of origin and makes an angle of `30^(0)` from positive direction of `X`-axis.

Answer

To find the equation of a line that intersects the X-axis at a distance of 2 units to the right of the origin and makes an angle of 30 degrees with the positive direction of the X-axis, we can follow these steps:

### Step 1: Identify the point of intersection
The line intersects the X-axis at a distance of 2 units to the right of the origin. Therefore, the point of intersection is:
\[ (2, 0) \]

### Step 2: Determine the slope of the line
The slope \( m \) of the line can be determined using the angle \( \theta \) it makes with the positive X-axis. The slope is given by:
\[ m = \tan(\theta) \]
Given \( \theta = 30^\circ \), we can calculate:
\[ m = \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

### Step 3: Use the point-slope form of the equation of a line
The point-slope form of the equation of a line is:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is a point on the line. Substituting \( (x_1, y_1) = (2, 0) \) and \( m = \frac{1}{\sqrt{3}} \):
\[ y - 0 = \frac{1}{\sqrt{3}}(x - 2) \]

### Step 4: Simplify the equation
Now, simplifying the equation:
\[ y = \frac{1}{\sqrt{3}}(x - 2) \]
\[ y = \frac{1}{\sqrt{3}}x - \frac{2}{\sqrt{3}} \]

### Step 5: Rearranging to standard form
To express the equation in standard form \( Ax + By + C = 0 \), we can multiply through by \( \sqrt{3} \) to eliminate the fraction:
\[ \sqrt{3}y = x - 2 \]
Rearranging gives:
\[ x - \sqrt{3}y - 2 = 0 \]

Thus, the final equation of the line is:
\[ x - \sqrt{3}y = 2 \]