Find equation 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 whose focus is at (1, -1), directrix is given by the line x - y - 3 = 0, and eccentricity is \( \frac{1}{2} \), we can follow these steps:

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

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

### Step 3: Calculate the distance from point \( P(x, y) \) to the focus \( F(1, -1) \)
Using the distance formula, we have:
\[ PF = \sqrt{(x - 1)^2 + (y + 1)^2} \]

### Step 4: Calculate the distance from point \( P(x, y) \) to the directrix
The distance from a point to a line \( Ax + By + C = 0 \) is given by:
\[ PD = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}} \]
For our directrix \( x - y - 3 = 0 \), we have \( A = 1, B = -1, C = -3 \). Thus,
\[ PD = \frac{|1 \cdot x - 1 \cdot y - 3|}{\sqrt{1^2 + (-1)^2}} = \frac{|x - y - 3|}{\sqrt{2}} \]

### Step 5: Set up the equation using eccentricity
Substituting the distances into the eccentricity formula:
\[ \frac{1}{2} = \frac{\sqrt{(x - 1)^2 + (y + 1)^2}}{\frac{|x - y - 3|}{\sqrt{2}}} \]

### Step 6: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ \sqrt{2} \cdot \sqrt{(x - 1)^2 + (y + 1)^2} = \frac{1}{2} |x - y - 3| \]

### Step 7: Square both sides to eliminate the square root
Squaring both sides results in:
\[ 2((x - 1)^2 + (y + 1)^2) = \frac{1}{4} (x - y - 3)^2 \]

### Step 8: Expand both sides
Expanding the left-hand 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-hand side:
\[ \frac{1}{4}(x - y - 3)^2 = \frac{1}{4}(x^2 - 2xy + y^2 - 6x + 6y + 9) \]

### Step 9: Combine and simplify
Setting both sides equal gives:
\[ 2x^2 + 2y^2 - 4x + 4y + 4 = \frac{1}{4}(x^2 - 2xy + y^2 - 6x + 6y + 9) \]
Multiply through by 4 to eliminate the fraction:
\[ 8x^2 + 8y^2 - 16x + 16y + 16 = x^2 - 2xy + y^2 - 6x + 6y + 9 \]

### Step 10: Rearranging the equation
Rearranging gives:
\[ 7x^2 + 7y^2 + 2xy - 10x + 10y + 7 = 0 \]

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