If 2x + y + k - 0 is a focal chord of `y^(2) + 4x` = 0 then k =

Options

A. 2
B. 4
C. `-4`
D. `-2`

Answer

To solve the problem, we need to find the value of \( k \) such that the line \( 2x + y + k = 0 \) is a focal chord of the parabola defined by the equation \( y^2 + 4x = 0 \).

### Step-by-step Solution:

1. **Identify the Parabola**:
The given equation of the parabola is \( y^2 + 4x = 0 \). We can rewrite this as:
\[ y^2 = -4x \]
This shows that it is a parabola that opens to the left.

2. **Find the Focus of the Parabola**:
The standard form of a parabola that opens to the left is \( y^2 = -4px \). Here, \( 4p = 4 \), so \( p = 1 \). The focus of this parabola is located at:
\[ (-p, 0) = (-1, 0) \]

3. **Substitute the Focus into the Focal Chord Equation**:
A focal chord passes through the focus of the parabola. The equation of the focal chord is given as:
\[ 2x + y + k = 0 \]
We will substitute the coordinates of the focus \((-1, 0)\) into this equation:
\[ 2(-1) + 0 + k = 0 \]

4. **Solve for k**:
Simplifying the equation:
\[ -2 + k = 0 \]
Adding 2 to both sides gives:
\[ k = 2 \]

5. **Conclusion**:
Thus, the value of \( k \) is:
\[ \boxed{2} \]