Question:
The equation of the directrix of the parabola $y^2 + 4y + 4x + 2 = 0$ is
The equation of the directrix of the parabola $y^2 + 4y + 4x + 2 = 0$ is
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Always complete the square first to identify $h,k,a$ correctly.
Updated On: Apr 30, 2026
- $x = -1$
- $x = 1$
- $x = \frac{3}{2}$
- $x = -\frac{3}{2}$
- $x = 2$
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The Correct Option is C
Solution and Explanation
Concept:
Standard form:
\[
(y-k)^2 = 4a(x-h), \text{Directrix: } x = h - a
\]
Step 1: Complete the square.
\[ y^2 + 4y + 4x + 2 = 0 \] \[ (y^2 + 4y) + 4x + 2 = 0 \] \[ (y+2)^2 - 4 + 4x + 2 = 0 \] \[ (y+2)^2 + 4x - 2 = 0 \]
Step 2: Convert to standard form.
\[ (y+2)^2 = -4(x - \tfrac{1}{2}) \]
Step 3: Identify parameters.
\[ h = \tfrac{1}{2}, a = -1 \]
Step 4: Directrix.
\[ x = h - a = \tfrac{1}{2} - (-1) = \tfrac{3}{2} \] \[ \boxed{x = \frac{3}{2}} \]
Step 1: Complete the square.
\[ y^2 + 4y + 4x + 2 = 0 \] \[ (y^2 + 4y) + 4x + 2 = 0 \] \[ (y+2)^2 - 4 + 4x + 2 = 0 \] \[ (y+2)^2 + 4x - 2 = 0 \]
Step 2: Convert to standard form.
\[ (y+2)^2 = -4(x - \tfrac{1}{2}) \]
Step 3: Identify parameters.
\[ h = \tfrac{1}{2}, a = -1 \]
Step 4: Directrix.
\[ x = h - a = \tfrac{1}{2} - (-1) = \tfrac{3}{2} \] \[ \boxed{x = \frac{3}{2}} \]
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