<p>The equation of the directrix of the parabola \(y^2 + 4y + 4x + 2 = 0\) is</p>
Step-by-Step Solution
Key Concept: Rewrite the parabola equation in standard form (y-k)² = 4p(x-h) by completing the square, then identify that the directrix is the vertical line x = h - p for a parabola opening horizontally.
<p><strong>Step 1:</strong> Complete the square for the given equation y² + 4y + 4x + 2 = 0</p><p>Rearrange: y² + 4y = -4x - 2</p><p><strong>Step 2:</strong> Complete the square on the left side: (y + 2)² - 4 = -4x - 2</p><p>(y + 2)² = -4x - 2 + 4 = -4x + 2</p><p><strong>Step 3:</strong> Factor out the coefficient of x: (y + 2)² = -4(x - 1/2)</p><p><strong>Step 4:</strong> This is in the form (y - k)² = 4p(x - h) where h = 1/2, k = -2, and 4p = -4, so p = -1</p><p><strong>Step 5:</strong> For a parabola (y - k)² = 4p(x - h), the directrix is x = h - p</p><p>Directrix: x = 1/2 - (-1) = 1/2 + 1 = 3/2</p><p>∴ Answer: <strong>x = 3/2</strong></p>
Correct Answer: B