The focus at `(1,1)` the directrix `x-y=3`.

Answer

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

### Step 1: Understand the Definition of a Parabola
A parabola is defined as the set of all points \( P \) such that the distance from \( P \) to the focus \( F \) is equal to the distance from \( P \) to the directrix \( D \).

### Step 2: Set Up the Problem
Let \( P(x, y) \) be a point on the parabola. The focus \( F \) is at \( (1, 1) \) and the directrix is given by the equation \( x - y = 3 \). We can rewrite the directrix in the standard form:
\[ x - y - 3 = 0 \]

### Step 3: Calculate the Distance from Point \( P \) to the Focus \( F \)
The distance \( P_F \) from point \( P(x, y) \) to the focus \( F(1, 1) \) is given by the distance formula:
\[ P_F = \sqrt{(x - 1)^2 + (y - 1)^2} \]

### Step 4: Calculate the Distance from Point \( P \) to the Directrix
The distance \( P_M \) from point \( P(x, y) \) to the directrix can be calculated using the formula for the distance from a point to a line \( Ax + By + C = 0 \):
\[ P_M = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Here, \( A = 1 \), \( B = -1 \), and \( C = -3 \). Thus,
\[ P_M = \frac{|1 \cdot x - 1 \cdot y - 3|}{\sqrt{1^2 + (-1)^2}} = \frac{|x - y - 3|}{\sqrt{2}} \]

### Step 5: Set the Distances Equal
According to the definition of the parabola, we have:
\[ P_F = P_M \]
Substituting the distances we found:
\[ \sqrt{(x - 1)^2 + (y - 1)^2} = \frac{|x - y - 3|}{\sqrt{2}} \]

### Step 6: Square Both Sides
Squaring both sides to eliminate the square root gives:
\[ (x - 1)^2 + (y - 1)^2 = \frac{(x - y - 3)^2}{2} \]

### Step 7: Multiply Through by 2
To eliminate the fraction, multiply both sides by 2:
\[ 2((x - 1)^2 + (y - 1)^2) = (x - y - 3)^2 \]

### Step 8: Expand Both Sides
Expanding the left side:
\[ 2((x^2 - 2x + 1) + (y^2 - 2y + 1)) = 2x^2 + 2y^2 - 4x - 4y + 4 \]
Expanding the right side:
\[ (x - y - 3)^2 = x^2 - 2xy + y^2 + 9 - 6x + 6y \]

### Step 9: Set the Expanded Forms Equal
Now we have:
\[ 2x^2 + 2y^2 - 4x - 4y + 4 = x^2 - 2xy + y^2 + 9 - 6x + 6y \]

### Step 10: Rearrange the Equation
Rearranging gives:
\[ 2x^2 + 2y^2 - x^2 - y^2 + 2xy + 2x - 10y - 5 = 0 \]
This simplifies to:
\[ x^2 + y^2 + 2xy + 2x - 10y - 5 = 0 \]

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