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

Options

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

Explanation

To find the equation of the parabola with a focus at (0, -3) and a directrix of y = 3, we can follow these steps:

### Step 1: Understand the properties of a parabola
A parabola is defined as the set of all points (P) that are equidistant from the focus (F) and the directrix (D). The distance from a point P(x, y) to the focus F(0, -3) must equal the perpendicular distance from P to the directrix y = 3.

### Step 2: Write the distance from point P to the focus
The distance \( Pf \) from point P(x, y) to the focus F(0, -3) can be calculated using the distance formula:
\[ Pf = \sqrt{(x - 0)^2 + (y + 3)^2} = \sqrt{x^2 + (y + 3)^2} \]

### Step 3: Write the distance from point P to the directrix
The directrix is the line y = 3. The perpendicular distance \( Ps \) from point P(x, y) to the directrix can be expressed as:
\[ Ps = |y - 3| \]

### Step 4: Set the distances equal to each other
For a parabola, we have:
\[ Pf = Ps \]
Thus,
\[ \sqrt{x^2 + (y + 3)^2} = |y - 3| \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ x^2 + (y + 3)^2 = (y - 3)^2 \]

### Step 6: Expand both sides
Expanding both sides:
- Left side:
\[ x^2 + (y^2 + 6y + 9) = x^2 + y^2 + 6y + 9 \]
- Right side:
\[ (y^2 - 6y + 9) \]

### Step 7: Set the expanded equations equal
Now we have:
\[ x^2 + y^2 + 6y + 9 = y^2 - 6y + 9 \]

### Step 8: Simplify the equation
Cancelling \( y^2 \) and \( 9 \) from both sides:
\[ x^2 + 6y = -6y \]
This simplifies to:
\[ x^2 = -12y \]

### Step 9: Write the final equation
Rearranging gives us the equation of the parabola:
\[ x^2 = -12y \]

### Conclusion
The equation of the parabola with focus (0, -3) and directrix y = 3 is:
\[ x^2 = -12y \]