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

Options

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

Answer

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

### Step 1: Understand the definition of a parabola
A parabola is defined as the set of all points \(P(x, y)\) that are equidistant from the focus and the directrix.

### Step 2: Set up the distance equations
Let \(P(h, k)\) be a point on the parabola. The distance from point \(P\) to the focus \(S(1, -1)\) is given by:
\[ PS = \sqrt{(h - 1)^2 + (k + 1)^2} \]

The distance from point \(P\) to the directrix \(x + y + 3 = 0\) can be calculated using the formula for the distance from a point to a line:
\[ PM = \frac{|h + k + 3|}{\sqrt{1^2 + 1^2}} = \frac{|h + k + 3|}{\sqrt{2}} \]

### Step 3: Set the distances equal
Since \(P\) is on the parabola, we set the distances equal:
\[ \sqrt{(h - 1)^2 + (k + 1)^2} = \frac{|h + k + 3|}{\sqrt{2}} \]

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

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

Expanding the right side:
\[ \frac{(h + k + 3)^2}{2} = \frac{h^2 + 2hk + k^2 + 6h + 6k + 9}{2} \]

### Step 6: Clear the fraction
Multiply both sides by 2 to eliminate the fraction:
\[ 2(h^2 + k^2 - 2h + 2k + 2) = h^2 + 2hk + k^2 + 6h + 6k + 9 \]

### Step 7: Rearrange the equation
This simplifies to:
\[ 2h^2 + 2k^2 - 4h + 4k + 4 = h^2 + 2hk + k^2 + 6h + 6k + 9 \]
Rearranging gives:
\[ h^2 + k^2 - 2hk - 10h - 2k - 5 = 0 \]

### Step 8: Substitute back to standard variables
Substituting \(h = x\) and \(k = y\), we get:
\[ x^2 + y^2 - 10x - 2y - 5 = 0 \]

### Final Step: Rearrange to standard form
This is the equation of the parabola in standard form.