The equation of the parabola with focus (0,0) and directrix x + y - 4 = 0 is

Options

A. `x^(2) + y^(2) - 2x y + 8x + 8y = 0 `
B. `x^(2) + y^(2) - 2xy + 8 x + 8 y - 16 = 0`
C. ` x^(2) + y^(2) + 8 x + 8y - 16 = 0 `
D. ` x^(2) + y^(2) + 2xy - 8x - 8y + 16 = 0 `

Answer

To find the equation of the parabola with focus at (0,0) and directrix given by the line \(x + y - 4 = 0\), we can follow these steps:

### Step 1: Understand the Definition of a Parabola
A parabola is defined as the set of all points (P) that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

### Step 2: Identify the Given Information
- Focus (F) = (0, 0)
- Directrix: \(x + y - 4 = 0\)

### Step 3: Set Up the Distance Equations
Let \(P(x, y)\) be any point on the parabola. The distance from point \(P\) to the focus \(F(0, 0)\) is given by:
\[ d(F, P) = \sqrt{(x - 0)^2 + (y - 0)^2} = \sqrt{x^2 + y^2} \]

The distance from point \(P\) to the directrix can be calculated using the formula for the distance from a point to a line \(Ax + By + C = 0\):
\[ d(P, \text{directrix}) = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}} \]
For the directrix \(x + y - 4 = 0\), we have:
- \(A = 1\), \(B = 1\), \(C = -4\)

Thus, the distance from point \(P\) to the directrix is:
\[ d(P, \text{directrix}) = \frac{|1 \cdot x + 1 \cdot y - 4|}{\sqrt{1^2 + 1^2}} = \frac{|x + y - 4|}{\sqrt{2}} \]

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

### Step 5: Square Both Sides
To eliminate the square root, we square both sides:
\[ x^2 + y^2 = \frac{(x + y - 4)^2}{2} \]

### Step 6: Multiply Through by 2
To clear the fraction, we multiply both sides by 2:
\[ 2(x^2 + y^2) = (x + y - 4)^2 \]

### Step 7: Expand the Right Side
Now, we expand the right side:
\[ 2(x^2 + y^2) = (x^2 + 2xy + y^2 - 8x - 8y + 16) \]

### Step 8: Rearrange the Equation
Now, we rearrange the equation:
\[ 2x^2 + 2y^2 = x^2 + y^2 + 2xy - 8x - 8y + 16 \]
\[ 2x^2 + 2y^2 - x^2 - y^2 - 2xy + 8x + 8y - 16 = 0 \]
\[ x^2 + y^2 - 2xy + 8x + 8y - 16 = 0 \]

### Final Equation
Thus, the equation of the parabola is:
\[ x^2 + y^2 - 2xy + 8x + 8y - 16 = 0 \]