If `(0,4)` and `(0, 2)` are respectively the vertex and focus of a parabola, then its equation is

Options

A. `x^2 +8y=32`
B. `y^2 +8x = 32`
C. `x^2 – 8 y = 32`
D. `y^2 - 8x=32`

Answer

To find the equation of the parabola with vertex at (0, 4) and focus at (0, 2), we can follow these steps:

### Step 1: Identify the vertex and focus
The vertex (V) is given as (0, 4) and the focus (F) is given as (0, 2).

### Step 2: Determine the orientation of the parabola
Since the focus is below the vertex, the parabola opens downward.

### Step 3: Find the distance 'a'
The distance 'a' between the vertex and the focus is calculated as:
\[ a = |y_{vertex} - y_{focus}| = |4 - 2| = 2 \]

### Step 4: Write the standard form of the equation
For a parabola that opens downward, the standard form of the equation is:
\[ (x - h)^2 = -4a(y - k) \]
where (h, k) is the vertex.

### Step 5: Substitute the values into the equation
Here, the vertex (h, k) is (0, 4) and 'a' is 2. Substituting these values into the equation gives:
\[ (x - 0)^2 = -4(2)(y - 4) \]
This simplifies to:
\[ x^2 = -8(y - 4) \]

### Step 6: Rearranging the equation
Now, we can rearrange this equation:
\[ x^2 = -8y + 32 \]
or
\[ x^2 + 8y = 32 \]

### Final Equation
Thus, the equation of the parabola is:
\[ x^2 + 8y = 32 \]