If the focus of parabola is at (0,-2) and its directrix is `y = 3`, then its equation is

Options

A. `x^2=-12y`
B. `x^2=12y`
C. `y^2=-12y`
D. `y^2=12x`

Answer

To find the equation of the parabola given its focus and directrix, we can follow these steps:

### Step 1: Identify the focus and directrix
The focus of the parabola is given as \( F(0, -2) \) and the directrix is given as \( y = 3 \).

### Step 2: Determine the vertex of the parabola
The vertex of the parabola lies midway between the focus and the directrix. The y-coordinate of the vertex can be found by averaging the y-coordinates of the focus and the directrix:
\[ y_v = \frac{(-2) + 3}{2} = \frac{1}{2} \]
Thus, the vertex \( V \) is at \( (0, \frac{1}{2}) \).

### Step 3: Determine the distance \( p \)
The distance \( p \) from the vertex to the focus (or from the vertex to the directrix) is given by:
\[ p = \text{Distance from vertex to focus} = \frac{1}{2} - (-2) = \frac{1}{2} + 2 = \frac{5}{2} \]
Since the focus is below the vertex, \( p \) is negative, so \( p = -\frac{5}{2} \).

### Step 4: Write the standard form of the parabola
The standard equation for a parabola that opens downwards is given by:
\[ (x - h)^2 = -4p(y - k) \]
where \( (h, k) \) is the vertex. Substituting \( h = 0 \), \( k = \frac{1}{2} \), and \( p = -\frac{5}{2} \):
\[ (x - 0)^2 = -4\left(-\frac{5}{2}\right)\left(y - \frac{1}{2}\right) \]
This simplifies to:
\[ x^2 = 10\left(y - \frac{1}{2}\right) \]

### Step 5: Expand and simplify the equation
Expanding the equation gives:
\[ x^2 = 10y - 5 \]
Rearranging this results in:
\[ x^2 - 10y + 5 = 0 \]

### Final Equation
Thus, the equation of the parabola is:
\[ x^2 - 10y + 5 = 0 \]

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