The equation of the set of all points which are equidistant from the point (0, 4) and the line `y = -4`

Options

A. `x^(2) = 16y`
B. `x^(2) =-16y`
C. `y^(2) =16x`
D. `y^(2) = -16x`

Answer

To find the equation of the set of all points that are equidistant from the point (0, 4) and the line \( y = -4 \), we can follow these steps:

### Step 1: Identify the Focus and Directrix
The point (0, 4) is the focus of the parabola, and the line \( y = -4 \) is the directrix.

### Step 2: Set Up the Distance Formula
For any point \( (x, y) \) on the parabola, the distance from the point to the focus (0, 4) is given by:
\[ d_1 = \sqrt{(x - 0)^2 + (y - 4)^2} = \sqrt{x^2 + (y - 4)^2} \]

The distance from the point \( (x, y) \) to the line \( y = -4 \) is the vertical distance, which can be calculated as:
\[ d_2 = |y - (-4)| = |y + 4| \]

### Step 3: Set the Distances Equal
Since the point is equidistant from the focus and the directrix, we have:
\[ \sqrt{x^2 + (y - 4)^2} = |y + 4| \]

### Step 4: Square Both Sides
To eliminate the square root, we square both sides:
\[ x^2 + (y - 4)^2 = (y + 4)^2 \]

### Step 5: Expand Both Sides
Expanding both sides gives:
\[ x^2 + (y^2 - 8y + 16) = (y^2 + 8y + 16) \]

### Step 6: Simplify the Equation
Now, we can simplify the equation:
\[ x^2 + y^2 - 8y + 16 = y^2 + 8y + 16 \]
Subtract \( y^2 + 16 \) from both sides:
\[ x^2 - 8y = 8y \]
Combine like terms:
\[ x^2 = 16y \]

### Step 7: Rearranging the Equation
This can be rearranged to the standard form of a parabola:
\[ x^2 = 16y \]

### Final Answer
The equation of the parabola is:
\[ x^2 = 16y \]

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