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

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

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 given equation is: \[ y^2 + 4x + 4y - 3 = 0. \] First, let's rearrange the terms to focus on completing the square for the $ y $ terms: \[ y^2 + 4y = -4x + 3. \] To complete the square for the $ y $ terms, we take the coefficient of $ y $, which is 4, divide it by 2 to get 2, and then square it to get 4. We add and subtract this square inside the equation: \[ (y^2 + 4y + 4) = -4x + 3 + 4. \] This simplifies to: \[ (y + 2)^2 = -4x + 7. \] Rearranging gives: \[ (y + 2)^2 = -4(x - \frac{7}{4}). \] This is now in the standard form of a parabola $(y - k)^2 = 4p(x - h)$, where the vertex is $(h, k)$ and the directrix is given by $x = h - p$. From the equation $(y + 2)^2 = -4(x - \frac{7}{4})$, we identify: - $h = \frac{7}{4}$, - $k = -2$, - $4p = -4$, so $p = -1$. The directrix of the parabola is given by: \[ x = h - p = \frac{7}{4} - (-1) = \frac{7}{4} + 1 = \frac{11}{4}. \] Multiplying through by 4 to clear the fraction, we get: \[ 4x = 11. \] 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 what the directrix of a parabola parallel to the y-axis should be. Therefore, it cannot be the directrix. 2. **Option 2: $11x = 4y$** - Similar to Option 1, this equation represents a line with a slope, not a vertical line. Thus, it cannot be the directrix of this parabola. 3. **Option 4: $4y = 11$** - This equation represents a horizontal line, which would be the directrix of a parabola that opens upwards or downwards, not sideways as in this case. Therefore, it is incorrect for this parabola. In conclusion, Option 3, $4x = 11$, is the correct representation of the directrix for the given parabola.

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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 given equation is:

$$ y^2 + 4x + 4y - 3 = 0. $$

First, let's rearrange the terms to focus on completing the square for the $ y $ terms:

$$ y^2 + 4y = -4x + 3. $$

To complete the square for the $ y $ terms, we take the coefficient of $ y $, which is 4, divide it by 2 to get 2, and then square it to get 4. We add and subtract this square inside the equation:

$$ (y^2 + 4y + 4) = -4x + 3 + 4. $$

This simplifies to:

$$ (y + 2)^2 = -4x + 7. $$

Rearranging gives:

$$ (y + 2)^2 = -4(x - \frac{7}{4}). $$

This is now in the standard form of a parabola $(y - k)^2 = 4p(x - h)$, where the vertex is $(h, k)$ and the directrix is given by $x = h - p$.

From the equation $(y + 2)^2 = -4(x - \frac{7}{4})$, we identify:

- $h = \frac{7}{4}$,
- $k = -2$,
- $4p = -4$, so $p = -1$.

The directrix of the parabola is given by:

$$ x = h - p = \frac{7}{4} - (-1) = \frac{7}{4} + 1 = \frac{11}{4}. $$

Multiplying through by 4 to clear the fraction, we get:

$$ 4x = 11. $$

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 what the directrix of a parabola parallel to the y-axis should be. Therefore, it cannot be the directrix.

2. **Option 2: $11x = 4y$** - Similar to Option 1, this equation represents a line with a slope, not a vertical line. Thus, it cannot be the directrix of this parabola.

3. **Option 4: $4y = 11$** - This equation represents a horizontal line, which would be the directrix of a parabola that opens upwards or downwards, not sideways as in this case. Therefore, it is incorrect for this parabola.

In conclusion, Option 3, $4x = 11$, is the correct representation of the directrix for the given parabola.