If question of the ellipse whose focus is `(1,-1),` then directrix the line `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)+7=0`
C. `7x^(2)+2xy+7y^(2)+10x-10y-7=0`
D. `"none of these"`

Explanation

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

### Step 1: Identify the given information
- Focus \( F(1, -1) \)
- Directrix \( x - y - 3 = 0 \)
- Eccentricity \( e = \frac{1}{2} \)

### Step 2: Write the relationship between the focus, directrix, and any point \( P(x, y) \) on the ellipse
By the definition of a conic section, we have:
\[ \frac{SP}{PM} = e \]
where \( SP \) is the distance from the focus to the point \( P \), and \( PM \) is the perpendicular distance from the point \( P \) to the directrix.

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

### Step 4: Calculate \( PM \)
To find \( PM \), we need the formula for the perpendicular distance from a point to a line. The line equation is \( Ax + By + C = 0 \), where \( A = 1, B = -1, C = -3 \). The formula for the distance from point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is:
\[ PM = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Substituting \( (x_0, y_0) = (x, y) \):
\[ PM = \frac{|x - y - 3|}{\sqrt{1^2 + (-1)^2}} = \frac{|x - y - 3|}{\sqrt{2}} \]

### Step 5: Set up the equation using the eccentricity
From the definition of the ellipse:
\[ \frac{SP}{PM} = \frac{1}{2} \]
Substituting the expressions for \( SP \) and \( PM \):
\[ \frac{\sqrt{(x - 1)^2 + (y + 1)^2}}{\frac{|x - y - 3|}{\sqrt{2}}} = \frac{1}{2} \]
Cross-multiplying gives:
\[ 2\sqrt{(x - 1)^2 + (y + 1)^2} = |x - y - 3|\sqrt{2} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides results in:
\[ 4((x - 1)^2 + (y + 1)^2) = 2(x - y - 3)^2 \]
Dividing both sides by 2:
\[ 2((x - 1)^2 + (y + 1)^2) = (x - y - 3)^2 \]

### Step 7: Expand both sides
Expanding the left side:
\[ 2((x - 1)^2 + (y + 1)^2) = 2[(x^2 - 2x + 1) + (y^2 + 2y + 1)] = 2x^2 - 4x + 2 + 2y^2 + 4y + 2 = 2x^2 + 2y^2 - 4x + 4y + 4 \]

Expanding the right side:
\[ (x - y - 3)^2 = x^2 - 2xy + y^2 - 6x + 6y + 9 \]

### Step 8: Set the equation
Setting both sides equal:
\[ 2x^2 + 2y^2 - 4x + 4y + 4 = x^2 - 2xy + y^2 - 6x + 6y + 9 \]

### Step 9: Rearranging the equation
Rearranging gives:
\[ 2x^2 - x^2 + 2y^2 - y^2 + 6x - 4x - 6y + 4y + 4 - 9 = 0 \]
This simplifies to:
\[ x^2 + y^2 + 2xy + 2x - 2y - 5 = 0 \]

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