Find the equation of the hyperbola whose one focus is `(1,1)` the corresponding directrix is `2x+y-1=0` and `e=sqrt(3)`.

Answer

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

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

### Step 2: Set up the coordinates of a point on the hyperbola
Let the coordinates of any point on the hyperbola be \((x, y)\).

### Step 3: Calculate the distance from the focus to the point
The distance from the focus (1, 1) to the point \((x, y)\) is given by:
\[ d_F = \sqrt{(x - 1)^2 + (y - 1)^2} \]

### Step 4: Calculate the perpendicular distance from the point to the directrix
The perpendicular distance from the point \((x, y)\) to the directrix \(2x + y - 1 = 0\) is given by:
\[ d_D = \frac{|2x + y - 1|}{\sqrt{2^2 + 1^2}} = \frac{|2x + y - 1|}{\sqrt{5}} \]

### Step 5: Use the definition of a hyperbola
According to the definition of a hyperbola, the distance from the focus to a point on the hyperbola is \(e\) times the distance from the point to the directrix:
\[ d_F = e \cdot d_D \]
Substituting the distances we calculated:
\[ \sqrt{(x - 1)^2 + (y - 1)^2} = \sqrt{3} \cdot \frac{|2x + y - 1|}{\sqrt{5}} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (x - 1)^2 + (y - 1)^2 = \frac{3}{5}(2x + y - 1)^2 \]

### Step 7: Expand both sides
Expanding the left 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 side:
\[ \frac{3}{5}(2x + y - 1)^2 = \frac{3}{5}(4x^2 + 4xy + y^2 - 4x - 2y + 1) \]
\[ = \frac{3}{5}(4x^2 + y^2 + 4xy - 4x - 2y + 1) \]

### Step 8: Combine and simplify
Now we can set the two sides equal to each other:
\[ x^2 + y^2 - 2x - 2y + 2 = \frac{3}{5}(4x^2 + y^2 + 4xy - 4x - 2y + 1) \]

### Step 9: Clear the fraction
Multiply through by 5 to eliminate the fraction:
\[ 5(x^2 + y^2 - 2x - 2y + 2) = 3(4x^2 + y^2 + 4xy - 4x - 2y + 1) \]

### Step 10: Expand and rearrange
Expanding both sides gives:
\[ 5x^2 + 5y^2 - 10x - 10y + 10 = 12x^2 + 3y^2 + 12xy - 12x - 6y + 3 \]

### Step 11: Move all terms to one side
Rearranging results in:
\[ 5x^2 + 5y^2 - 10x - 10y + 10 - 12x^2 - 3y^2 - 12xy + 12x + 6y - 3 = 0 \]
This simplifies to:
\[ -7x^2 + 2y^2 - 12xy + 2x - 4y + 7 = 0 \]

### Final Equation
Thus, the equation of the hyperbola is:
\[ 7x^2 - 2y^2 + 12xy - 2x + 4y - 7 = 0 \]