What is the equation of the straight line cutting off an intercept 2 from the negative direction of y-axis and inclined at `30^(@)` with the positive direction of x-axis ?

Options

A. `x - 2 sqrt3 y - 3sqrt2 = 0`
B. `x + 2 sqrt3 y - 3sqrt2 = 0`
C. `x + sqrt3y - 2sqrt3 = 0`
D. `x - sqrt3y - 2sqrt3 = 0`

Answer

To find the equation of the straight line that cuts off an intercept of 2 from the negative direction of the y-axis and is inclined at \(30^\circ\) with the positive direction of the x-axis, we can follow these steps:

### Step 1: Identify the y-intercept
The problem states that the line cuts off an intercept of 2 from the negative direction of the y-axis. This means that the y-intercept \(c\) is \(-2\).

### Step 2: Find the slope of the line
The slope \(m\) of the line can be calculated using the angle of inclination. The formula for the slope in terms of the angle \(\theta\) is:
\[ m = \tan(\theta) \]
Given that \(\theta = 30^\circ\):
\[ m = \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

### Step 3: Write the equation of the line
The equation of a straight line in slope-intercept form is given by:
\[ y = mx + c \]
Substituting the values of \(m\) and \(c\):
\[ y = \frac{1}{\sqrt{3}}x - 2 \]

### Step 4: Rearranging the equation
To express the equation in standard form \(Ax + By + C = 0\), we can rearrange the equation:
\[ y + 2 = \frac{1}{\sqrt{3}}x \]
Multiplying through by \(\sqrt{3}\) to eliminate the fraction:
\[ \sqrt{3}y + 2\sqrt{3} = x \]
Rearranging gives:
\[ x - \sqrt{3}y - 2\sqrt{3} = 0 \]

### Final Equation
Thus, the equation of the line is:
\[ x - \sqrt{3}y - 2\sqrt{3} = 0 \]