Find the equation of the hyperbola whose focus is `(1,2),` directrix ` 2x +y = 1 and ` eccentricity ` sqrt3.`

Answer

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

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

### Step 2: Set up the distance formula
Let \( P(x, y) \) be a point on the hyperbola. According to the definition of a hyperbola, the distance from point \( P \) to the focus \( S \) is equal to \( e \) times the distance from point \( P \) to the directrix.

1. Distance from \( P \) to the focus \( S \):
\[ SP = \sqrt{(x - 1)^2 + (y - 2)^2} \]

2. Distance from \( P \) to the directrix \( 2x + y - 1 = 0 \):
\[ PM = \frac{|2x + y - 1|}{\sqrt{2^2 + 1^2}} = \frac{|2x + y - 1|}{\sqrt{5}} \]

### Step 3: Set up the equation based on the definition of hyperbola
According to the definition:
\[ SP = e \cdot PM \]
Substituting the distances we found:
\[ \sqrt{(x - 1)^2 + (y - 2)^2} = \sqrt{3} \cdot \frac{|2x + y - 1|}{\sqrt{5}} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (x - 1)^2 + (y - 2)^2 = \frac{3}{5} (2x + y - 1)^2 \]

### Step 5: Expand both sides
1. Left-hand side:
\[ (x - 1)^2 + (y - 2)^2 = (x^2 - 2x + 1) + (y^2 - 4y + 4) = x^2 + y^2 - 2x - 4y + 5 \]

2. Right-hand side:
\[ \frac{3}{5} (2x + y - 1)^2 = \frac{3}{5} (4x^2 + 4xy - 4x + y^2 - 2y + 1) = \frac{3}{5}(4x^2 + y^2 + 4xy - 4x - 2y + 1) \]

### Step 6: Combine and simplify the equation
Now, we will equate both sides:
\[ x^2 + y^2 - 2x - 4y + 5 = \frac{3}{5}(4x^2 + 4xy - 4x + y^2 - 2y + 1) \]

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

### Step 7: Rearranging the terms
After expanding and rearranging, we will get a quadratic equation in terms of \( x \) and \( y \):
\[ 7x^2 - 2y^2 + 12xy - 2x + 14y - 22 = 0 \]

### Final Equation
Thus, the equation of the hyperbola is:
\[ 7x^2 - 2y^2 + 12xy - 2x + 14y - 22 = 0 \]