Find the equation of the line whose inclination is ` 60^(@)` and passes through `(-2, 4)`

Options

A. ` sqrt""3 x + y + 6 = 0`
B. ` 3 x + y + 6 = 0 `
C. ` x -3 y + 6 = 0`
D. None of these

Answer

To find the equation of the line with an inclination of \(60^\circ\) that passes through the point \((-2, 4)\), we can follow these steps:

### Step 1: Determine the slope of the line
The slope \(m\) of a line can be found using the tangent of the angle of inclination:
\[ m = \tan(\theta) \]
For an inclination of \(60^\circ\):
\[ m = \tan(60^\circ) = \sqrt{3} \]

### 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, \((-2, 4)\) gives us \(x_1 = -2\) and \(y_1 = 4\).

### Step 3: Substitute the values into the point-slope form
Substituting \(m = \sqrt{3}\), \(x_1 = -2\), and \(y_1 = 4\) into the equation:
\[ y - 4 = \sqrt{3}(x - (-2)) \]
This simplifies to:
\[ y - 4 = \sqrt{3}(x + 2) \]

### Step 4: Distribute the slope on the right-hand side
Distributing \(\sqrt{3}\):
\[ y - 4 = \sqrt{3}x + 2\sqrt{3} \]

### Step 5: Rearrange the equation to standard form
To express the equation in standard form, we can move all terms to one side:
\[ y - \sqrt{3}x - 2\sqrt{3} + 4 = 0 \]
This can be rearranged as:
\[ -\sqrt{3}x + y + (4 - 2\sqrt{3}) = 0 \]
Or, multiplying through by \(-1\) for a more standard appearance:
\[ \sqrt{3}x - y + (2\sqrt{3} - 4) = 0 \]

### Final Equation
Thus, the equation of the line is:
\[ \sqrt{3}x - y + (2\sqrt{3} - 4) = 0 \]