The equation of directrix of a hyperbola is `3x + 4y + 8 = 0` The focus of the hyperbola is `(1, 1)`. If eccentricity of the hyperbola is 2 and the length of conjugate axis is k then [k], where [] represents the greatest integer function, is equal to ________ .

Answer

To solve the problem step by step, we need to find the length of the conjugate axis \( k \) of the hyperbola given the focus, directrix, and eccentricity.

### Step 1: Identify the given information
- Directrix: \( 3x + 4y + 8 = 0 \)
- Focus: \( (1, 1) \)
- Eccentricity \( e = 2 \)

### Step 2: Calculate the perpendicular distance from the focus to the directrix
The formula for the perpendicular distance \( d \) from a point \( (x_1, y_1) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
For the directrix \( 3x + 4y + 8 = 0 \), we have:
- \( A = 3 \)
- \( B = 4 \)
- \( C = 8 \)

Substituting the focus \( (1, 1) \):
\[ d = \frac{|3(1) + 4(1) + 8|}{\sqrt{3^2 + 4^2}} = \frac{|3 + 4 + 8|}{\sqrt{9 + 16}} = \frac{15}{5} = 3 \]

### Step 3: Relate the distance to the eccentricity
For a hyperbola, the relationship between the distance from the focus to the directrix and the semi-major axis \( a \) is given by:
\[ d = \frac{a}{e} \]
Substituting \( d = 3 \) and \( e = 2 \):
\[ 3 = \frac{a}{2} \implies a = 3 \times 2 = 6 \]

### Step 4: Use the relationship between \( a \), \( b \), and \( e \)
We know that for hyperbolas:
\[ e^2 = 1 + \frac{b^2}{a^2} \]
Substituting \( e = 2 \) and \( a = 6 \):
\[ 4 = 1 + \frac{b^2}{6^2} \implies 4 = 1 + \frac{b^2}{36} \]
This simplifies to:
\[ 3 = \frac{b^2}{36} \implies b^2 = 3 \times 36 = 108 \implies b = \sqrt{108} = 6\sqrt{3} \]

### Step 5: Calculate the length of the conjugate axis
The length of the conjugate axis \( k \) is given by:
\[ k = 2b = 2(6\sqrt{3}) = 12\sqrt{3} \]

### Step 6: Find the greatest integer value of \( k \)
To find \( [k] \), we need to calculate \( 12\sqrt{3} \). Since \( \sqrt{3} \approx 1.732 \):
\[ k \approx 12 \times 1.732 \approx 20.784 \]
Thus, the greatest integer function gives:
\[ [k] = 20 \]

### Final Answer
The greatest integer value of \( k \) is \( \boxed{20} \).