Find the equation of parabola whose
Vertex is (-1,-2) latus rectum is 4 and axis is parallel to y axis

Answer

To find the equation of the parabola with the given conditions, we can follow these steps:

### Step 1: Identify the standard form of the parabola
Since the axis of the parabola is parallel to the y-axis, the standard form of the equation of the parabola is:
\[ (x - x_1)^2 = 4a(y - y_1) \]
where \((x_1, y_1)\) is the vertex of the parabola.

### Step 2: Substitute the vertex coordinates
The vertex is given as \((-1, -2)\). Therefore, we have:
\[ x_1 = -1 \quad \text{and} \quad y_1 = -2 \]
Substituting these values into the standard form gives:
\[ (x + 1)^2 = 4a(y + 2) \]

### Step 3: Use the length of the latus rectum to find \(a\)
The length of the latus rectum is given as 4. The length of the latus rectum is related to \(a\) by the formula:
\[ \text{Length of latus rectum} = 4a \]
Setting this equal to the given length:
\[ 4a = 4 \]
From this, we can solve for \(a\):
\[ a = 1 \]

### Step 4: Substitute \(a\) back into the equation
Now that we have \(a\), we can substitute it back into the equation:
\[ (x + 1)^2 = 4 \cdot 1 \cdot (y + 2) \]
This simplifies to:
\[ (x + 1)^2 = 4(y + 2) \]

### Step 5: Final equation of the parabola
Thus, the equation of the parabola is:
\[ (x + 1)^2 = 4(y + 2) \]

### Summary
The equation of the parabola whose vertex is \((-1, -2)\), with a latus rectum of 4 and an axis parallel to the y-axis is:
\[ (x + 1)^2 = 4(y + 2) \]