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 will follow these steps:

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

### Step 2: Convert the directrix equation to the standard form
The equation of the directrix can be rewritten as:
\[ 4x - 3y - 3 = 0 \]
This is in the form \( Ax + By + C = 0 \) where \( A = 4 \), \( B = -3 \), and \( C = -3 \).

### Step 3: Calculate the distance from the focus to the directrix
The distance \( d \) from the focus (3, 0) to the directrix can be calculated using the formula for the distance from a point to a line:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Substituting \( (x_1, y_1) = (3, 0) \):
\[ d = \frac{|4(3) - 3(0) - 3|}{\sqrt{4^2 + (-3)^2}} = \frac{|12 - 3|}{\sqrt{16 + 9}} = \frac{9}{5} = 1.8 \]

### Step 4: Relate the distances using the eccentricity
For a hyperbola, the relationship between the distance from the focus to a point on the hyperbola (P) and the distance from that point to the directrix (D) is given by:
\[ PF = e \cdot PD \]
Where \( PF \) is the distance from the focus to the point on the hyperbola, and \( PD \) is the distance from the point to the directrix.

Given that \( e = \frac{5}{4} \) and \( PD = d = 1.8 \):
\[ PF = \frac{5}{4} \cdot PD = \frac{5}{4} \cdot 1.8 = \frac{5 \cdot 1.8}{4} = \frac{9}{4} = 2.25 \]

### Step 5: Set up the equation of the hyperbola
The standard form of the hyperbola is:
\[ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \]
Where \( (h, k) \) is the center of the hyperbola, \( a \) is the distance from the center to the vertices along the x-axis, and \( b \) is the distance from the center to the vertices along the y-axis.

### Step 6: Find the center and values of a and b
Since the focus is at (3, 0), we can assume the center is at (h, k) = (3, 0) for simplicity. The distance from the focus to the vertices is \( c = PF = 2.25 \).

Using the relationship \( c^2 = a^2 + b^2 \) and knowing \( e = \frac{c}{a} \):
\[ \frac{5}{4} = \frac{c}{a} \implies c = \frac{5}{4}a \]
Substituting \( c = 2.25 \):
\[ 2.25 = \frac{5}{4}a \implies a = \frac{2.25 \cdot 4}{5} = 1.8 \]

Now, we can find \( b \):
\[ c^2 = a^2 + b^2 \implies (2.25)^2 = (1.8)^2 + b^2 \]
Calculating:
\[ 5.0625 = 3.24 + b^2 \implies b^2 = 5.0625 - 3.24 = 1.8225 \]

### Step 7: Write the final equation of the hyperbola
Substituting \( a^2 \) and \( b^2 \) into the hyperbola equation:
\[ \frac{(x - 3)^2}{(1.8)^2} - \frac{y^2}{(1.8225)} = 1 \]
This simplifies to:
\[ \frac{(x - 3)^2}{3.24} - \frac{y^2}{1.8225} = 1 \]

### Final Answer
The equation of the hyperbola is:
\[ \frac{(x - 3)^2}{3.24} - \frac{y^2}{1.8225} = 1 \]