The equation of a line which passes through `(2,3)` and makes an angle of `30^(@)` with the positive direction of x-axis is

Options

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

Answer

To find the equation of a line that passes through the point (2, 3) and makes an angle of 30 degrees with the positive direction of the x-axis, we can follow these steps:

### Step 1: Find the slope of the line
The slope \( m \) of a line that makes an angle \( \theta \) with the positive direction of the x-axis is given by:
\[ m = \tan(\theta) \]
For \( \theta = 30^\circ \):
\[ m = \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

### Step 2: Use the point-slope form of the equation of a line
The point-slope form of the equation of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Where \( (x_1, y_1) \) is the point the line passes through. Here, \( (x_1, y_1) = (2, 3) \) and \( m = \frac{1}{\sqrt{3}} \).

Substituting the values into the equation:
\[ y - 3 = \frac{1}{\sqrt{3}}(x - 2) \]

### Step 3: Simplify the equation
Now, we can simplify the equation:
\[ y - 3 = \frac{1}{\sqrt{3}}x - \frac{2}{\sqrt{3}} \]
Adding 3 to both sides:
\[ y = \frac{1}{\sqrt{3}}x - \frac{2}{\sqrt{3}} + 3 \]
To combine the constants, we can express 3 in terms of a common denominator:
\[ 3 = \frac{3\sqrt{3}}{\sqrt{3}} \]
Thus, we have:
\[ y = \frac{1}{\sqrt{3}}x + \left(\frac{3\sqrt{3}}{\sqrt{3}} - \frac{2}{\sqrt{3}}\right) \]
\[ y = \frac{1}{\sqrt{3}}x + \frac{3\sqrt{3} - 2}{\sqrt{3}} \]

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

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