Directrix of the parabola $y^{2}+4x+4y-3=0$ is equal to:

Question

Accepted Answer

To determine the directrix of the given parabola $ y^2 + 4x + 4y - 3 = 0 $, we first need to rewrite the equation in the standard form of a parabola. The standard form for a parabola that opens sideways is $(y - k)^2 = 4p(x - h)$.

Let's start by rearranging and completing the square for the $ y $ terms:

1. Begin with the given equation:
   $$
   y^2 + 4x + 4y - 3 = 0
   $$

2. Rearrange the terms to group the $ y $ terms together:
   $$
   y^2 + 4y = -4x + 3
   $$

3. Complete the square for the $ y $ terms:
   - Take half of the coefficient of $ y $, which is 4, divide by 2 to get 2, and then square it to get 4.
   - Add and subtract 4 inside the equation:
   $$
   (y^2 + 4y + 4) = -4x + 3 + 4
   $$

4. This simplifies to:
   $$
   (y + 2)^2 = -4x + 7
   $$

5. Rearrange to match the standard form:
   $$
   (y + 2)^2 = -4(x - \frac{7}{4})
   $$

From this equation, we can see that the parabola opens to the left because the coefficient of $(x - h)$ is negative. The vertex of the parabola is $\left(\frac{7}{4}, -2\right)$, and $4p = -4$, so $p = -1$.

The directrix of a parabola that opens horizontally is given by the equation $x = h - p$. Here, $h = \frac{7}{4}$ and $p = -1$, so the directrix is:
$$
x = \frac{7}{4} + 1 = \frac{11}{4}
$$

This can be rewritten as:
$$
4x = 11
$$

Thus, the directrix of the parabola is $4x = 11$, which corresponds to Option 3.

**Explanation for Incorrect Options:**

1. **Option 1: $4x = 11y$**
   - This equation represents a line with a slope, not a vertical line, which is required for the directrix of a horizontally opening parabola.

2. **Option 2: $11x = 4y$**
   - Similar to Option 1, this equation represents a line with a slope, not a vertical line.

3. **Option 4: $4y = 11$**
   - This equation represents a horizontal line, which would be the directrix for a vertically opening parabola, not a horizontally opening one.

Therefore, Option 3, $4x = 11$, is the correct answer as it correctly represents the vertical line that serves as the directrix for the given parabola.