Equation of the directrix of the parabola `y^(2)+4x+2=0` is

Options

A. `x = -1`
B. `x=1`
C. `x= -3//2`
D. `x=1//2`

Answer

To find the equation of the directrix of the parabola given by the equation \( y^2 + 4x + 2 = 0 \), we will follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the equation to isolate the terms involving \( x \):
\[ y^2 + 4x + 2 = 0 \implies 4x = -y^2 - 2 \implies x = -\frac{1}{4}y^2 - \frac{1}{2} \]

### Step 2: Identify the standard form of the parabola
The standard form of a parabola that opens to the left is given by:
\[ y^2 = 4a(x - h) \]
where \( (h, k) \) is the vertex of the parabola and \( a \) is the distance from the vertex to the focus (and also to the directrix).

### Step 3: Rewrite the equation in standard form
From the rearranged equation, we can express it in the standard form:
\[ y^2 = -4\left(\frac{1}{4}\right)\left(x + \frac{1}{2}\right) \]
Here, we can see that \( 4a = -4 \) implies \( a = -1 \) and the vertex \( (h, k) = \left(-\frac{1}{2}, 0\right) \).

### Step 4: Find the directrix
The equation of the directrix for a parabola that opens to the left is given by:
\[ x = h + a \]
Substituting the values we found:
\[ x = -\frac{1}{2} + (-1) = -\frac{1}{2} - 1 = -\frac{3}{2} \]

### Conclusion
Thus, the equation of the directrix of the parabola \( y^2 + 4x + 2 = 0 \) is:
\[ \boxed{x = -\frac{3}{2}} \]