Find the equation of a line passing through the point `(-2,0)` and makes an angle of `(2pi)/(3)` from the positive direction of `X-`axis.

Answer

To find the equation of the line passing through the point \((-2, 0)\) and making an angle of \(\frac{2\pi}{3}\) (or \(120^\circ\)) with the positive direction of the X-axis, we can follow these steps:

### Step 1: Determine the slope of the line
The slope \(m\) of the line can be found using the tangent of the angle it makes with the positive X-axis. The formula for the slope in terms of the angle \(\theta\) is:
\[ m = \tan(\theta) \]
Here, \(\theta = \frac{2\pi}{3}\) or \(120^\circ\).

Calculating the slope:
\[ m = \tan\left(\frac{2\pi}{3}\right) = \tan(120^\circ) \]
Using the identity:
\[ \tan(120^\circ) = -\cot(30^\circ) = -\frac{1}{\tan(30^\circ)} = -\frac{1}{\frac{1}{\sqrt{3}}} = -\sqrt{3} \]
Thus, the slope \(m = -\sqrt{3}\).

### 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 a point on the line. Here, \((x_1, y_1) = (-2, 0)\) and \(m = -\sqrt{3}\).

Substituting these values into the equation:
\[ y - 0 = -\sqrt{3}(x - (-2)) \]
This simplifies to:
\[ y = -\sqrt{3}(x + 2) \]

### Step 3: Rearranging to standard form
Expanding the equation:
\[ y = -\sqrt{3}x - 2\sqrt{3} \]
To express this in standard form \(Ax + By + C = 0\), we can rearrange it:
\[ \sqrt{3}x + y + 2\sqrt{3} = 0 \]

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