the equation of the ellipse whose foucs is S(1,-1) directrix the line `x-y-3=0` and eccentricity` 1//2 `, is

Options

A. `7x^(2)+ 2xy + 7y^(2) - 10 x + 10 y + 7 =0`
B. `7x^(2)+ 2xy + 7y^(2) +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 \( S(1, -1) \), directrix \( x - y - 3 = 0 \), and eccentricity \( e = \frac{1}{2} \), we can follow these steps:

### Step 1: Understand the definition of an ellipse
An ellipse can be defined as the set of points \( P(x, y) \) such that the distance from \( P \) to the focus \( S \) is equal to \( e \) times the perpendicular distance from \( P \) to the directrix.

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

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

### Step 4: Set up the equation using the definition of the ellipse
According to the definition:
\[ SP = e \cdot PM \]
Substituting the values we have:
\[ \sqrt{(x - 1)^2 + (y + 1)^2} = \frac{1}{2} \cdot \frac{|x - y - 3|}{\sqrt{2}} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (x - 1)^2 + (y + 1)^2 = \frac{1}{4} \cdot \frac{(x - y - 3)^2}{2} \]
This simplifies to:
\[ 2((x - 1)^2 + (y + 1)^2) = \frac{1}{4}(x - y - 3)^2 \]

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

### Step 7: Set both sides equal and simplify
Setting both sides equal:
\[ 2x^2 + 2y^2 - 4x + 4y + 4 = \frac{1}{4}x^2 - \frac{1}{2}xy + \frac{1}{4}y^2 - \frac{3}{2}x + \frac{3}{2}y + \frac{9}{4} \]
Multiplying through by 4 to eliminate the fraction:
\[ 8x^2 + 8y^2 - 16x + 16y + 16 = x^2 - 2xy + y^2 - 6x + 6y + 9 \]

### Step 8: Rearranging and combining like terms
Rearranging gives:
\[ 7x^2 + 7y^2 + 2xy - 10x + 10y + 7 = 0 \]

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