Equation of the rectangular hyperbola whose focus is `(1,-1)` and the corresponding directrix is `x-y+1=0`

Options

A. `x^(2)-y^(2)=1`
B. `xy=1`
C. `2xy-4x+4y+1=0`
D. `2xy+4x-4y-1=0`

Explanation

To find the equation of the rectangular hyperbola whose focus is at (1, -1) and whose directrix is given by the equation \(x - y + 1 = 0\), we will follow these steps:

### Step 1: Understand the properties of a rectangular hyperbola
A rectangular hyperbola has an eccentricity \(e = \sqrt{2}\). The general formula for the distance from a point \((x, y)\) to the focus \((x_0, y_0)\) is given by:
\[ PS = \sqrt{(x - x_0)^2 + (y - y_0)^2} \]
where \(P\) is the point on the hyperbola and \(S\) is the focus.

### Step 2: Find the distance from the point to the directrix
The distance \(PM\) from a point \((x, y)\) to the directrix \(Ax + By + C = 0\) is given by:
\[ PM = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}} \]
For the directrix \(x - y + 1 = 0\), we have \(A = 1\), \(B = -1\), and \(C = 1\). Thus, the distance becomes:
\[ PM = \frac{|x - y + 1|}{\sqrt{1^2 + (-1)^2}} = \frac{|x - y + 1|}{\sqrt{2}} \]

### Step 3: Set up the equation using the definition of eccentricity
For a hyperbola, the relationship between the distances to the focus and the directrix is given by:
\[ \frac{PS}{PM} = e \]
Substituting the values we have:
\[ \frac{\sqrt{(x - 1)^2 + (y + 1)^2}}{\frac{|x - y + 1|}{\sqrt{2}}} = \sqrt{2} \]

### Step 4: Simplify the equation
Cross-multiplying gives:
\[ \sqrt{(x - 1)^2 + (y + 1)^2} = \sqrt{2} \cdot \frac{|x - y + 1|}{\sqrt{2}} \]
This simplifies to:
\[ \sqrt{(x - 1)^2 + (y + 1)^2} = |x - y + 1| \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides results in:
\[ (x - 1)^2 + (y + 1)^2 = (x - y + 1)^2 \]

### Step 6: Expand both sides
Expanding the left-hand side:
\[ (x^2 - 2x + 1) + (y^2 + 2y + 1) = x^2 + y^2 - 2x + 2y + 1 \]
The left-hand side becomes:
\[ x^2 + y^2 - 2x + 2y + 2 \]

Expanding the right-hand side:
\[ (x - y + 1)^2 = x^2 - 2xy + y^2 + 2x - 2y + 1 \]

### Step 7: Set the equation
Setting both expansions equal gives:
\[ x^2 + y^2 - 2x + 2y + 2 = x^2 - 2xy + y^2 + 2x - 2y + 1 \]

### Step 8: Simplify the equation
Cancelling \(x^2\) and \(y^2\) from both sides:
\[ -2x + 2y + 2 = -2xy + 2x - 2y + 1 \]
Rearranging gives:
\[ 2xy - 4x + 4y + 1 = 0 \]

### Step 9: Final form of the hyperbola equation
Thus, the equation of the rectangular hyperbola is:
\[ 2xy - 4x + 4y + 1 = 0 \]