The focus at `(-2,-1)` and the latus rectum joins the points `(-2,2)` and `(-2,-4)`.

Answer

To find the equation of the parabola with the given focus and latus rectum endpoints, we can follow these steps:

### Step 1: Identify the Coordinates
The focus of the parabola is given as \( F(-2, -1) \). The endpoints of the latus rectum are given as \( (-2, 2) \) and \( (-2, -4) \).

### Step 2: Determine the Orientation of the Parabola
Since the x-coordinates of the focus and the endpoints of the latus rectum are the same (all are \( x = -2 \)), the parabola opens either upwards or downwards. Given that the focus is below the latus rectum endpoints, the parabola opens upwards.

### Step 3: Find the Vertex
The vertex \( V \) of the parabola lies on the line of symmetry, which is the vertical line through the focus. The y-coordinate of the vertex can be found as the midpoint of the y-coordinates of the endpoints of the latus rectum:
\[ y_V = \frac{2 + (-4)}{2} = \frac{-2}{2} = -1 \]
Thus, the vertex \( V \) is at \( (-2, -1) \).

### Step 4: Calculate the Length of the Latus Rectum
The length of the latus rectum is the distance between the two endpoints:
\[ \text{Length} = 2 - (-4) = 6 \]
Since the length of the latus rectum is \( 4p \), we can set up the equation:
\[ 4p = 6 \implies p = \frac{6}{4} = \frac{3}{2} \]

### Step 5: Write the Equation of the Parabola
The standard form of the equation of a parabola that opens upwards is:
\[ (x - h)^2 = 4p(y - k) \]
Where \( (h, k) \) is the vertex. Substituting \( h = -2 \), \( k = -1 \), and \( p = \frac{3}{2} \):
\[ (x + 2)^2 = 6(y + 1) \]

### Final Equation
Thus, the equation of the parabola is:
\[ (x + 2)^2 = 6(y + 1) \]

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