Question

The equation of the directrix of the parabola $y^2 + 4y + 4x + 2 = 0$ is ______.

• A. $x = -1$
• B. $x = 1$
• C. $x = -\frac{3}{2}$
• D. $x = \frac{3}{2}$

Hint:

For a shifted horizontal parabola $(y-k)^2 = 4a(x-h)$ (opening right), the directrix is $x = h - a$. For $(y-k)^2 = -4a(x-h)$ (opening left), the directrix is $x = h + a$. Always draw a quick sketch to confirm which direction the directrix is relative to the vertex!

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Convert the general quadratic equation into the standard form $(y-k)^2 = -4a(x-h)$.

Step 2: Formula Application:
Complete the square for $y$: $y^2 + 4y + 4 = -4x - 2 + 4$ $(y+2)^2 = -4x + 2 = -4(x - 1/2)$.

Step 3: Explanation:
Here, $4a = 4 \implies a = 1$. The parabola opens to the left. The vertex is $(h, k) = (1/2, -2)$. For a parabola $(y-k)^2 = -4a(x-h)$, the directrix is $x = h + a$. Directrix: $x = 1/2 + 1 = 3/2$.

Step 4: Final Answer:
The equation of the directrix is $x = 3/2$.