Obtain the equation of the ellipse whose focus is the point (-1, 1), and the corresponding directrix is the line `x-y+3=0`, and the eccentricity is `1/2`.

Answer

To find the equation of the ellipse with the given focus, directrix, and eccentricity, we will follow these steps:

### Step 1: Identify the given information
- Focus (F) = (-1, 1)
- Directrix (D): x - y + 3 = 0
- Eccentricity (e) = 1/2

### Step 2: Write the formula for the distances
The definition of eccentricity for an ellipse is given by the formula:
\[ e = \frac{PF}{PD} \]
where:
- \( PF \) is the distance from a point \( P(h, k) \) on the ellipse to the focus \( F \).
- \( PD \) is the distance from the point \( P(h, k) \) to the directrix \( D \).

### Step 3: Calculate \( PF \)
Using the distance formula, the distance from point \( P(h, k) \) to the focus \( F(-1, 1) \) is:
\[ PF = \sqrt{(h + 1)^2 + (k - 1)^2} \]
Expanding this, we get:
\[ PF = \sqrt{h^2 + 2h + 1 + k^2 - 2k + 1} = \sqrt{h^2 + k^2 + 2h - 2k + 2} \]

### Step 4: Calculate \( PD \)
The distance from point \( P(h, k) \) to the directrix \( D: x - y + 3 = 0 \) can be calculated using the formula for the distance from a point to a line:
\[ PD = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Here, \( A = 1, B = -1, C = 3 \), and \( (x_1, y_1) = (h, k) \):
\[ PD = \frac{|h - k + 3|}{\sqrt{1^2 + (-1)^2}} = \frac{|h - k + 3|}{\sqrt{2}} \]

### Step 5: Set up the equation using eccentricity
Using the eccentricity definition:
\[ \frac{PF}{PD} = \frac{1}{2} \]
Substituting the expressions for \( PF \) and \( PD \):
\[ \frac{\sqrt{h^2 + k^2 + 2h - 2k + 2}}{\frac{|h - k + 3|}{\sqrt{2}}} = \frac{1}{2} \]
Cross-multiplying gives:
\[ 2\sqrt{h^2 + k^2 + 2h - 2k + 2} = |h - k + 3|\sqrt{2} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides:
\[ 4(h^2 + k^2 + 2h - 2k + 2) = 2(h - k + 3)^2 \]
Expanding both sides:
\[ 4h^2 + 4k^2 + 8h - 8k + 8 = 2(h^2 - 2hk + k^2 + 6h - 6k + 9) \]
\[ 4h^2 + 4k^2 + 8h - 8k + 8 = 2h^2 - 4hk + 2k^2 + 12h - 12k + 18 \]

### Step 7: Rearranging the equation
Bringing all terms to one side:
\[ 4h^2 - 2h^2 + 4k^2 - 2k^2 + 8h - 12h - 8k + 12k + 8 - 18 + 4hk = 0 \]
This simplifies to:
\[ 2h^2 + 2k^2 - 4h + 4k - 10 + 4hk = 0 \]
Dividing through by 2:
\[ h^2 + k^2 + 2hk - 2h + 2k - 5 = 0 \]

### Step 8: Replace \( h \) and \( k \) with \( x \) and \( y \)
Substituting \( h = x \) and \( k = y \):
\[ x^2 + y^2 + 2xy - 2x + 2y - 5 = 0 \]

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