Find the equation of the hyperbola whose one focus is `(-1, 1)` , eccentricity = 3 and the equation of the corresponding directrix is `x - y + 3 = 0 ` .

Answer

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

### Step 1: Identify the given information
- Focus \( S = (-1, 1) \)
- Eccentricity \( e = 3 \)
- Directrix equation: \( x - y + 3 = 0 \)

### Step 2: Set up the distance equations
For a point \( P(h, k) \) on the hyperbola, the relationship between the distances from the focus and the directrix is given by:
\[ PS = e \cdot PM \]
where \( PS \) is the distance from the point \( P \) to the focus \( S \) and \( PM \) is the perpendicular distance from the point \( P \) to the directrix.

### Step 3: Calculate the distance \( PS \)
The distance \( PS \) from the point \( P(h, k) \) to the focus \( S(-1, 1) \) is given by:
\[ PS = \sqrt{(h + 1)^2 + (k - 1)^2} \]

### Step 4: Calculate the perpendicular distance \( PM \)
The directrix is given by the equation \( x - y + 3 = 0 \). The perpendicular distance \( PM \) from the point \( P(h, k) \) to the directrix can be calculated using the formula:
\[ PM = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
where \( A = 1, B = -1, C = 3 \). Therefore,
\[ PM = \frac{|h - k + 3|}{\sqrt{1^2 + (-1)^2}} = \frac{|h - k + 3|}{\sqrt{2}} \]

### Step 5: Set up the equation using eccentricity
Substituting \( PS \) and \( PM \) into the equation \( PS = e \cdot PM \):
\[ \sqrt{(h + 1)^2 + (k - 1)^2} = 3 \cdot \frac{|h - k + 3|}{\sqrt{2}} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (h + 1)^2 + (k - 1)^2 = \frac{9(h - k + 3)^2}{2} \]

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

### Step 8: Combine and simplify the equation
Combining both sides gives:
\[ h^2 + k^2 + 2h - 2k + 2 = \frac{9}{2}(h^2 - 2hk + k^2 + 6h - 6k + 9) \]
Multiply through by 2 to eliminate the fraction:
\[ 2h^2 + 2k^2 + 4h - 4k + 4 = 9(h^2 - 2hk + k^2 + 6h - 6k + 9) \]

### Step 9: Rearrange the equation
Rearranging gives:
\[ 2h^2 + 2k^2 + 4h - 4k + 4 - 9h^2 + 18hk - 9k^2 - 54h + 54k - 81 = 0 \]
Combine like terms:
\[ -7h^2 - 7k^2 + 18hk - 50h + 50k - 77 = 0 \]

### Step 10: Replace \( h \) and \( k \) with \( x \) and \( y \)
Thus, the equation of the hyperbola is:
\[ 7x^2 + 7y^2 - 18xy + 50x - 50y + 77 = 0 \]

### Final Equation
The equation of the hyperbola is:
\[ 7x^2 + 7y^2 - 18xy + 50x - 50y + 77 = 0 \]

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