Find the length of the major axis of the ellipse whose focus is (1,-1) , corresponding directrix is the line `x - y - 3 = 0 ` and eccentricity is `(1)/(2)`

Answer

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

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

### Step 2: Write the equation of the directrix
The equation of the directrix can be rearranged to the form:
\[ x - y = 3 \]
This means that for any point \( P(h, k) \) on the ellipse, the distance from \( P \) to the directrix can be calculated.

### Step 3: Use the definition of eccentricity
The definition of eccentricity for an ellipse states:
\[ e = \frac{PS}{PM} \]
where \( PS \) is the distance from point \( P \) to the focus \( S \) and \( PM \) is the distance from point \( P \) to the directrix.

### Step 4: Calculate distances
1. **Distance \( PS \)**:
\[ PS = \sqrt{(h - 1)^2 + (k + 1)^2} \]

2. **Distance \( PM \)**:
The distance from point \( P(h, k) \) to the line \( x - y - 3 = 0 \) is given by the formula:
\[ PM = \frac{|x_1 - y_1 - 3|}{\sqrt{1^2 + (-1)^2}} = \frac{|h - k - 3|}{\sqrt{2}} \]

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

### Step 6: Square both sides to eliminate the square root
Squaring both sides results in:
\[ (h - k - 3)^2 = 8((h - 1)^2 + (k + 1)^2) \]

### Step 7: Expand both sides
Expanding the left side:
\[ (h - k - 3)^2 = h^2 - 2hk + k^2 - 6h + 6k + 9 \]
Expanding the right side:
\[ 8((h - 1)^2 + (k + 1)^2) = 8(h^2 - 2h + 1 + k^2 + 2k + 1) = 8h^2 + 8k^2 - 16h + 16k + 16 \]

### Step 8: Combine like terms
Setting the two expansions equal:
\[ h^2 - 2hk + k^2 - 6h + 6k + 9 = 8h^2 + 8k^2 - 16h + 16k + 16 \]
Rearranging gives:
\[ -7h^2 - 7k^2 + 2hk + 10h - 10k - 7 = 0 \]

### Step 9: Simplify the equation
Dividing through by -1:
\[ 7h^2 + 7k^2 - 2hk - 10h + 10k + 7 = 0 \]

### Step 10: Identify the lengths of the axes
The standard form of the ellipse is:
\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]
where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

From the eccentricity \( e = \frac{c}{a} \), where \( c \) is the distance from the center to the focus, we can find:
\[ c = ae \Rightarrow c = a \cdot \frac{1}{2} \]
Since \( c^2 = a^2 - b^2 \), we can find \( a \) and \( b \) and thus the length of the major axis \( 2a \).

### Final Calculation
Given \( e = \frac{1}{2} \), we can deduce that:
\[ a = 2c \Rightarrow a = 2 \cdot \frac{1}{2} = 1 \]
Thus, the length of the major axis is:
\[ 2a = 2 \cdot 1 = 2 \]

### Conclusion
The length of the major axis of the ellipse is \( 2 \).