Find the equation of hyperbola whose eccentricity is `5//4` , whose focus is `(3, 0)` and whose directrix is `4x - 3y = 3`.

Answer

To find the equation of the hyperbola with the given parameters, we can follow these steps:

### Step 1: Identify the Given Information
We are given:
- Eccentricity \( e = \frac{5}{4} \)
- Focus \( F(3, 0) \)
- Directrix \( 4x - 3y = 3 \)

### Step 2: Find the Distance from the Focus to a Point on the Hyperbola
Let \( P(x, y) \) be a point on the hyperbola. The distance from the point \( P \) to the focus \( F(3, 0) \) is given by:
\[ d_F = \sqrt{(x - 3)^2 + y^2} \]

### Step 3: Find the Distance from the Point to the Directrix
The distance from the point \( P(x, y) \) to the directrix \( 4x - 3y - 3 = 0 \) can be calculated using the formula for the distance from a point to a line:
\[ d_D = \frac{|4x - 3y - 3|}{\sqrt{4^2 + (-3)^2}} = \frac{|4x - 3y - 3|}{5} \]

### Step 4: Set Up the Relationship Using Eccentricity
For a hyperbola, the relationship between the distances is given by:
\[ d_F = e \cdot d_D \]
Substituting the values we have:
\[ \sqrt{(x - 3)^2 + y^2} = \frac{5}{4} \cdot \frac{|4x - 3y - 3|}{5} \]
This simplifies to:
\[ \sqrt{(x - 3)^2 + y^2} = \frac{1}{4} |4x - 3y - 3| \]

### Step 5: Square Both Sides
Squaring both sides to eliminate the square root gives:
\[ (x - 3)^2 + y^2 = \left(\frac{1}{4} (4x - 3y - 3)\right)^2 \]
Expanding both sides:
\[ (x - 3)^2 + y^2 = \frac{1}{16} (4x - 3y - 3)^2 \]

### Step 6: Expand and Simplify
Expanding the left side:
\[ (x^2 - 6x + 9 + y^2) = \frac{1}{16} (16x^2 - 24xy + 9y^2 - 24x + 24y + 9) \]
Multiply through by 16 to eliminate the fraction:
\[ 16(x^2 - 6x + 9 + y^2) = 16x^2 - 24xy + 9y^2 - 24x + 24y + 9 \]

### Step 7: Rearranging the Equation
Rearranging gives:
\[ 16x^2 - 96x + 144 + 16y^2 = 16x^2 - 24xy + 9y^2 - 24x + 24y + 9 \]
Cancelling \( 16x^2 \) from both sides:
\[ -96x + 144 + 16y^2 = -24xy + 9y^2 - 24x + 24y + 9 \]
Rearranging further:
\[ 24xy + 7y^2 - 72x + 135 = 0 \]

### Final Equation of the Hyperbola
Thus, the equation of the hyperbola is:
\[ 24xy + 7y^2 - 72x + 135 = 0 \]