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

Options

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

Answer

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

### Step 1: Understand the definition of a parabola
A parabola is defined as the set of all points \(P(x, y)\) such that the distance from \(P\) to the focus \(S(0, 0)\) is equal to the perpendicular distance from \(P\) to the directrix.

### Step 2: Identify the focus and directrix
- Focus \(S = (0, 0)\)
- Directrix: \(x + y = 4\)

### Step 3: Write the distance from point \(P(x, y)\) to the focus
The distance \(SP\) from point \(P(x, y)\) to the focus \(S(0, 0)\) is given by:
\[ SP = \sqrt{(x - 0)^2 + (y - 0)^2} = \sqrt{x^2 + y^2} \]

### Step 4: Write the perpendicular distance from point \(P(x, y)\) to the directrix
The distance \(PM\) from point \(P(x, y)\) to the line \(x + y - 4 = 0\) can be calculated using the formula for the distance from a point to a line:
\[ PM = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
where \(A = 1\), \(B = 1\), and \(C = -4\). Thus, we have:
\[ PM = \frac{|1 \cdot x + 1 \cdot y - 4|}{\sqrt{1^2 + 1^2}} = \frac{|x + y - 4|}{\sqrt{2}} \]

### Step 5: Set the distances equal
According to the definition of a parabola, we set \(SP = PM\):
\[ \sqrt{x^2 + y^2} = \frac{|x + y - 4|}{\sqrt{2}} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ x^2 + y^2 = \frac{(x + y - 4)^2}{2} \]

### Step 7: Multiply through by 2 to eliminate the fraction
\[ 2(x^2 + y^2) = (x + y - 4)^2 \]

### Step 8: Expand the right-hand side
Expanding the right side:
\[ (x + y - 4)^2 = x^2 + y^2 + 8x + 8y + 16 - 2xy \]
Thus, we have:
\[ 2x^2 + 2y^2 = x^2 + y^2 + 8x + 8y + 16 - 2xy \]

### Step 9: Rearrange the equation
Rearranging gives:
\[ 2x^2 + 2y^2 - x^2 - y^2 + 2xy - 8x - 8y - 16 = 0 \]
This simplifies to:
\[ x^2 + y^2 + 2xy - 8x - 8y - 16 = 0 \]

### Step 10: Final equation of the parabola
Thus, the equation of the parabola is:
\[ x^2 + y^2 + 2xy - 8x - 8y - 16 = 0 \]