The equation of the ellipse with focus (-1,1) directrix `x-y+3=0` and eccentricity `(1)/(2)` is

Options

A. `7x^2+2xy+7y^2+10x+10y+7=0`
B. `7x^2+2xy+7y^2+10x-10y+7=0`
C. `7x^2+2xy+7y^2+10x-10y-7=0`
D. None of these

Answer

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

### Step 1: Understand the properties of the ellipse
The property of an ellipse states that for any point \( P(x, y) \) on the ellipse, the distance from \( P \) to the focus \( S(-1, 1) \) is equal to the eccentricity \( e \) multiplied by the perpendicular distance from \( P \) to the directrix.

### Step 2: Write down the distances
1. **Distance from point \( P(x, y) \) to the focus \( S(-1, 1) \)**:
\[ PS = \sqrt{(x + 1)^2 + (y - 1)^2} \]

2. **Perpendicular distance from point \( P(x, y) \) to the directrix \( x - y + 3 = 0 \)**:
The formula for the perpendicular distance from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ \text{Distance} = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Here, \( A = 1, B = -1, C = 3 \), so the distance \( PM \) is:
\[ PM = \frac{|x - y + 3|}{\sqrt{1^2 + (-1)^2}} = \frac{|x - y + 3|}{\sqrt{2}} \]

### Step 3: Set up the equation using the property of the ellipse
Using the property of the ellipse:
\[ PS = e \cdot PM \]
Substituting the values we have:
\[ \sqrt{(x + 1)^2 + (y - 1)^2} = \frac{1}{2} \cdot \frac{|x - y + 3|}{\sqrt{2}} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (x + 1)^2 + (y - 1)^2 = \frac{1}{4} \cdot \frac{(x - y + 3)^2}{2} \]
Simplifying this, we have:
\[ (x + 1)^2 + (y - 1)^2 = \frac{1}{8} (x - y + 3)^2 \]

### Step 5: Expand both sides
1. **Left side**:
\[ (x + 1)^2 + (y - 1)^2 = x^2 + 2x + 1 + y^2 - 2y + 1 = x^2 + y^2 + 2x - 2y + 2 \]

2. **Right side**:
\[ \frac{1}{8} (x - y + 3)^2 = \frac{1}{8} (x^2 - 2xy + y^2 + 6x - 6y + 9) \]

### Step 6: Combine and simplify
Setting both sides equal:
\[ x^2 + y^2 + 2x - 2y + 2 = \frac{1}{8} (x^2 - 2xy + y^2 + 6x - 6y + 9) \]
Multiplying through by 8 to eliminate the fraction:
\[ 8(x^2 + y^2 + 2x - 2y + 2) = x^2 - 2xy + y^2 + 6x - 6y + 9 \]

### Step 7: Rearranging the equation
Bringing all terms to one side:
\[ 8x^2 + 8y^2 + 16x - 16y + 16 - x^2 + 2xy - y^2 - 6x + 6y - 9 = 0 \]
Combining like terms:
\[ (8x^2 - x^2) + (8y^2 - y^2) + (16x - 6x) + (2xy) + (-16y + 6y) + (16 - 9) = 0 \]
This simplifies to:
\[ 7x^2 + 2xy + 7y^2 - 10y + 7 = 0 \]

### Final Equation
Thus, the equation of the ellipse is:
\[ 7x^2 + 2xy + 7y^2 - 10y + 7 = 0 \]