The equation of the ellipse whose focus is (1,-1), directrix `x-y-3=0` and eccentricity equals `(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

Answer

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

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

### Step 2: Write the general equation of the conic section
The equation of a conic section with focus \((x_0, y_0)\), directrix \(Lx + My + N = 0\), and eccentricity \(e\) is given by:

\[ (x - x_0)^2 + (y - y_0)^2 = e^2 \cdot \frac{(Lx + My + N)^2}{L^2 + M^2} \]

### Step 3: Substitute the values into the equation
Here, \(x_0 = 1\), \(y_0 = -1\), \(L = 1\), \(M = -1\), and \(N = -3\).

Substituting these values into the equation:

\[ (x - 1)^2 + (y + 1)^2 = \left(\frac{1}{2}\right)^2 \cdot \frac{(1 \cdot x - 1 \cdot y - 3)^2}{1^2 + (-1)^2} \]

### Step 4: Simplify the equation
Calculating \(e^2\):

\[ e^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

Calculating \(L^2 + M^2\):

\[ L^2 + M^2 = 1^2 + (-1)^2 = 1 + 1 = 2 \]

Now substituting these into the equation:

\[ (x - 1)^2 + (y + 1)^2 = \frac{1}{4} \cdot \frac{(x - y - 3)^2}{2} \]

This simplifies to:

\[ (x - 1)^2 + (y + 1)^2 = \frac{1}{8} (x - y - 3)^2 \]

### Step 5: Expand both sides
Expanding the left-hand side:

\[ (x - 1)^2 + (y + 1)^2 = x^2 - 2x + 1 + y^2 + 2y + 1 = x^2 + y^2 - 2x + 2y + 2 \]

Expanding the right-hand side:

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

### Step 6: Combine and simplify
Now we equate both sides:

\[ x^2 + y^2 - 2x + 2y + 2 = \frac{1}{8} (x^2 - 2xy + y^2 - 6x + 6y + 9) \]

Multiplying through by 8 to eliminate the fraction:

\[ 8(x^2 + y^2 - 2x + 2y + 2) = x^2 - 2xy + y^2 - 6x + 6y + 9 \]

Expanding the left side:

\[ 8x^2 + 8y^2 - 16x + 16y + 16 = x^2 - 2xy + y^2 - 6x + 6y + 9 \]

### Step 7: Rearranging the equation
Now, we rearrange all terms to one side:

\[ 8x^2 - x^2 + 8y^2 - y^2 + 2xy - 16x + 6x + 16y - 6y + 16 - 9 = 0 \]

This simplifies to:

\[ 7x^2 + 7y^2 + 2xy - 10x + 10y + 7 = 0 \]

### Final Equation
Thus, the equation of the ellipse is:

\[ 7x^2 + 7y^2 + 2xy - 10x + 10y + 7 = 0 \]