Find the vertex and length of latus rectum of the parabola `x^(2)=-4(y-a)`.

Answer

To find the vertex and length of the latus rectum of the parabola given by the equation \( x^2 = -4(y - a) \), we can follow these steps:

### Step 1: Identify the standard form of the parabola
The given equation can be rewritten in the standard form of a parabola that opens downwards. The standard form is:
\[ x^2 = -4p(y - k) \]
where \((0, k)\) is the vertex and \(4p\) is the length of the latus rectum.

### Step 2: Compare the given equation with the standard form
From the equation \( x^2 = -4(y - a) \), we can see that:
- \(k = a\)
- \(4p = 4\)

### Step 3: Find the vertex
The vertex of the parabola is given by the point \((0, k)\). From our comparison:
\[ k = a \implies \text{Vertex} = (0, a) \]

### Step 4: Find the value of \(p\)
From the equation \(4p = 4\), we can solve for \(p\):
\[ p = 1 \]

### Step 5: Find the length of the latus rectum
The length of the latus rectum is given by the formula \(4p\):
\[ \text{Length of latus rectum} = 4p = 4 \times 1 = 4 \]

### Final Results
- **Vertex**: \((0, a)\)
- **Length of Latus Rectum**: \(4\)

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