Find the equation to the hyperbola whose eccentiricity is 2, whose focus is `(2,0)` and whose directrix is `x - y =0.`

Answer

To find the equation of the hyperbola with the given conditions, we will follow these steps:

### Step 1: Identify the parameters of the hyperbola
We know:
- Eccentricity \( e = 2 \)
- Focus \( F(2, 0) \)
- Directrix \( x - y = 0 \) or \( y = x \)

### Step 2: Use the definition of hyperbola
The definition of a hyperbola states that for any point \( P(x, y) \) on the hyperbola, the ratio of the distance from \( P \) to the focus and the distance from \( P \) to the directrix is equal to the eccentricity \( e \).

Mathematically, this can be expressed as:
\[ \frac{d(P, F)}{d(P, D)} = e \]
where \( d(P, F) \) is the distance from point \( P \) to the focus \( F \) and \( d(P, D) \) is the distance from point \( P \) to the directrix \( D \).

### Step 3: Calculate the distances
1. **Distance from \( P(x, y) \) to the focus \( F(2, 0) \)**:
\[ d(P, F) = \sqrt{(x - 2)^2 + (y - 0)^2} = \sqrt{(x - 2)^2 + y^2} \]

2. **Distance from \( P(x, y) \) to the line \( x - y = 0 \)**:
The distance from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For the line \( x - y = 0 \) (which can be rewritten as \( 1x - 1y + 0 = 0 \)):
\[ d(P, D) = \frac{|1 \cdot x - 1 \cdot y + 0|}{\sqrt{1^2 + (-1)^2}} = \frac{|x - y|}{\sqrt{2}} \]

### Step 4: Set up the equation
Using the definition of the hyperbola:
\[ \frac{\sqrt{(x - 2)^2 + y^2}}{\frac{|x - y|}{\sqrt{2}}} = 2 \]

### Step 5: Cross-multiply and simplify
Cross-multiplying gives:
\[ \sqrt{(x - 2)^2 + y^2} = 2 \cdot \frac{|x - y|}{\sqrt{2}} \]
Squaring both sides:
\[ (x - 2)^2 + y^2 = 2 \cdot 2 \cdot \frac{(x - y)^2}{2} \]
This simplifies to:
\[ (x - 2)^2 + y^2 = 2(x - y)^2 \]

### Step 6: Expand and rearrange
Expanding both sides:
\[ (x^2 - 4x + 4) + y^2 = 2(x^2 - 2xy + y^2) \]
This simplifies to:
\[ x^2 - 4x + 4 + y^2 = 2x^2 - 4xy + 2y^2 \]
Rearranging gives:
\[ 0 = x^2 - 4x + 4 + y^2 - 2x^2 + 4xy - 2y^2 \]
\[ 0 = -x^2 + 4xy - y^2 - 4x + 4 \]
Multiplying through by -1:
\[ x^2 - 4xy + y^2 + 4x - 4 = 0 \]

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

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