Find the equation to the conic section whose focus is (1, -1), eccentricity is `(1/2)` and the directrix is the line `x-y=3`. Is the conic section an ellipse?

Answer

To find the equation of the conic section with the given focus, eccentricity, and directrix, we will follow these steps:

### Step 1: Understand the given information
- Focus (F) = (1, -1)
- Eccentricity (e) = 1/2
- Directrix: x - y = 3

### Step 2: Set up the distance formulas
Let P(h, k) be any point on the conic section. The definition of a conic section states that the eccentricity \( e \) is the ratio of the distance from the point P to the focus (F) and the distance from the point P to the directrix.

1. **Distance from P to F (focus)**:
\[ d_f = \sqrt{(h - 1)^2 + (k + 1)^2} \]

2. **Distance from P to the directrix**:
The distance from a point (h, k) to the line Ax + By + C = 0 is given by:
\[ d_d = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}} \]
For the line x - y - 3 = 0, we have A = 1, B = -1, and C = -3. Thus,
\[ d_d = \frac{|h - k - 3|}{\sqrt{1^2 + (-1)^2}} = \frac{|h - k - 3|}{\sqrt{2}} \]

### Step 3: Apply the definition of eccentricity
Using the definition of eccentricity:
\[ e = \frac{d_f}{d_d} \]
Substituting the distances:
\[ \frac{1}{2} = \frac{\sqrt{(h - 1)^2 + (k + 1)^2}}{\frac{|h - k - 3|}{\sqrt{2}}} \]

### Step 4: Cross-multiply and simplify
Cross-multiplying gives:
\[ \sqrt{2} \cdot \sqrt{(h - 1)^2 + (k + 1)^2} = \frac{1}{2} |h - k - 3| \]
Squaring both sides:
\[ 2((h - 1)^2 + (k + 1)^2) = \frac{1}{4} (h - k - 3)^2 \]

### Step 5: Expand both sides
Expanding the left side:
\[ 2((h - 1)^2 + (k + 1)^2) = 2(h^2 - 2h + 1 + k^2 + 2k + 1) = 2h^2 + 2k^2 - 4h + 4k + 4 \]

Expanding the right side:
\[ \frac{1}{4} (h - k - 3)^2 = \frac{1}{4}(h^2 - 2hk + k^2 - 6h + 6k + 9) \]

### Step 6: Combine and simplify
Combining the equations:
\[ 2h^2 + 2k^2 - 4h + 4k + 4 = \frac{1}{4}(h^2 - 2hk + k^2 - 6h + 6k + 9) \]
Multiplying through by 4 to eliminate the fraction:
\[ 8h^2 + 8k^2 - 16h + 16k + 16 = h^2 - 2hk + k^2 - 6h + 6k + 9 \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ 7h^2 + 7k^2 + 2hk - 10h + 10k + 7 = 0 \]

### Step 8: Replace variables
Replacing h with x and k with y gives:
\[ 7x^2 + 7y^2 + 2xy - 10x + 10y + 7 = 0 \]

### Conclusion
This is the equation of the conic section. Since the eccentricity \( e = \frac{1}{2} \) is between 0 and 1, the conic section is indeed an ellipse.