The equation of a straight line drawn through the focus of the parabola `y^(2) = - 4x` at an angle of `120^(@)` to x axis is

Options

A. `y+sqrt(3)(x-1)=0`
B. `y-sqrt(3)(x-1)=1`
C. `y+sqrt(3)(x+1)=0`
D. `y-sqrt(3)(x+1)=0`

Answer

To find the equation of a straight line drawn through the focus of the parabola \(y^2 = -4x\) at an angle of \(120^\circ\) to the x-axis, we can follow these steps:

### Step 1: Identify the focus of the parabola
The given parabola is in the form \(y^2 = -4ax\). Here, \(4a = 4\), which gives us \(a = 1\). The focus of the parabola \(y^2 = -4x\) is located at the point \((-a, 0)\). Thus, the focus is:
\[ \text{Focus} = (-1, 0) \]

### Step 2: Determine the slope of the line
The slope \(m\) of a line that makes an angle \(\theta\) with the positive x-axis is given by:
\[ m = \tan(\theta) \]
For \(\theta = 120^\circ\):
\[ m = \tan(120^\circ) = \tan(180^\circ - 60^\circ) = -\tan(60^\circ) = -\sqrt{3} \]

### Step 3: Use the point-slope form to write the equation of the line
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Here, \((x_1, y_1) = (-1, 0)\) and \(m = -\sqrt{3}\). Substituting these values into the equation:
\[ y - 0 = -\sqrt{3}(x - (-1)) \]
This simplifies to:
\[ y = -\sqrt{3}(x + 1) \]

### Step 4: Rearranging the equation
To express the equation in standard form, we can rearrange it:
\[ y + \sqrt{3}x + \sqrt{3} = 0 \]
Thus, the final equation of the line is:
\[ \sqrt{3}x + y + \sqrt{3} = 0 \]

### Final Answer
The equation of the straight line drawn through the focus of the parabola \(y^2 = -4x\) at an angle of \(120^\circ\) to the x-axis is:
\[ \sqrt{3}x + y + \sqrt{3} = 0 \]

---