The equation of the parabola with focus `(1,-1)` and directrix `x + y + 3 =0` is

Options

A. `x ^(2) + y^(2) -10 x - 2y-2xy -5=0`
B. `x ^(2) + y ^(2) + 10 x -2y - 2xy-5=0`
C. `x ^(2) + y ^(2) + 10x + 2y -2xy -5=0`
D. `x ^(2) + y ^(2) + 10x + 2y + 2x y -5=0`

Answer

To find the equation of the parabola with focus at \( (1, -1) \) and directrix given by the line \( x + y + 3 = 0 \), we will follow these steps:

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

### Step 2: Write the equation of the parabola
The general definition of a parabola is that it is the set of all points \( (x, y) \) that are equidistant from the focus and the directrix.

The distance from a point \( (x, y) \) to the focus \( (1, -1) \) is given by:
\[ d_1 = \sqrt{(x - 1)^2 + (y + 1)^2} \]

The distance from the point \( (x, y) \) to the directrix \( x + y + 3 = 0 \) can be calculated using the formula for the distance from a point to a line \( Ax + By + C = 0 \):
\[ d_2 = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Here, \( A = 1, B = 1, C = 3 \), and \( (x_0, y_0) = (x, y) \):
\[ d_2 = \frac{|x + y + 3|}{\sqrt{1^2 + 1^2}} = \frac{|x + y + 3|}{\sqrt{2}} \]

### Step 3: Set the distances equal
Since the parabola is defined by the condition that these distances are equal, we have:
\[ \sqrt{(x - 1)^2 + (y + 1)^2} = \frac{|x + y + 3|}{\sqrt{2}} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (x - 1)^2 + (y + 1)^2 = \frac{(x + y + 3)^2}{2} \]

### Step 5: Expand both sides
Expanding the left side:
\[ (x - 1)^2 + (y + 1)^2 = (x^2 - 2x + 1) + (y^2 + 2y + 1) = x^2 + y^2 - 2x + 2y + 2 \]

Expanding the right side:
\[ \frac{(x + y + 3)^2}{2} = \frac{x^2 + 2xy + y^2 + 6x + 6y + 9}{2} \]

### Step 6: Multiply through by 2 to eliminate the fraction
Multiplying the entire equation by 2 gives:
\[ 2(x^2 + y^2 - 2x + 2y + 2) = x^2 + 2xy + y^2 + 6x + 6y + 9 \]

### Step 7: Simplify the equation
This simplifies to:
\[ 2x^2 + 2y^2 - 4x + 4y + 4 = x^2 + 2xy + y^2 + 6x + 6y + 9 \]
Rearranging gives:
\[ x^2 + y^2 - 2xy - 10x - 2y - 5 = 0 \]

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