To find the vertex, focus, directrix, and latus rectum of the parabola given by the equation \( y^2 + 4x + 4y - 3 = 0 \), we will follow these steps:
### Step 1: Rearranging the equation
Start by rearranging the equation to isolate the \( y^2 \) term:
\[
y^2 + 4y = -4x + 3
\]
### Step 2: Completing the square
Next, we complete the square for the \( y \) terms:
\[
y^2 + 4y + 4 = -4x + 3 + 4
\]
This simplifies to:
\[
(y + 2)^2 = -4x + 7
\]
or
\[
(y + 2)^2 = -4(x - \frac{7}{4})
\]
### Step 3: Identifying the vertex
From the equation \( (y - k)^2 = 4p(x - h) \), we can identify the vertex \((h, k)\):
- Here, \( h = \frac{7}{4} \) and \( k = -2 \).
- Thus, the vertex is:
\[
\text{Vertex} = \left( \frac{7}{4}, -2 \right)
\]
### Step 4: Finding the focus
The value of \( p \) (the distance from the vertex to the focus) is given by \( p = -1 \) (since \( 4p = -4 \)). The focus is located at:
\[
\text{Focus} = \left( h + p, k \right) = \left( \frac{7}{4} - 1, -2 \right) = \left( \frac{3}{4}, -2 \right)
\]
### Step 5: Finding the directrix
The directrix is a vertical line given by:
\[
x = h - p = \frac{7}{4} + 1 = \frac{11}{4}
\]
### Step 6: Finding the latus rectum
The length of the latus rectum is given by \( |4p| \):
\[
\text{Length of Latus Rectum} = |4 \cdot (-1)| = 4
\]
The endpoints of the latus rectum can be found by moving \( 2 \) units up and down from the focus:
\[
\text{Endpoints} = \left( \frac{3}{4}, -2 + 2 \right) \text{ and } \left( \frac{3}{4}, -2 - 2 \right) = \left( \frac{3}{4}, 0 \right) \text{ and } \left( \frac{3}{4}, -4 \right)
\]
### Summary of Results
- **Vertex**: \( \left( \frac{7}{4}, -2 \right) \)
- **Focus**: \( \left( \frac{3}{4}, -2 \right) \)
- **Directrix**: \( x = \frac{11}{4} \)
- **Length of Latus Rectum**: \( 4 \) (Endpoints: \( \left( \frac{3}{4}, 0 \right) \) and \( \left( \frac{3}{4}, -4 \right) \))
