If `kx+3y=1, 2x+y+5=0` are conjugate liens w.r.t the circle `x^(2)+y^(2)-2x-4y-4=0` then k=

Options

A. 3
B. 4
C. 2
D. 1

Answer

To solve the problem, we need to determine the value of \( k \) such that the lines \( kx + 3y = 1 \) and \( 2x + y + 5 = 0 \) are conjugate with respect to the circle given by the equation \( x^2 + y^2 - 2x - 4y - 4 = 0 \).

### Step-by-Step Solution:

1. **Rewrite the equations of the lines:**
- Line 1: \( kx + 3y - 1 = 0 \) (Let \( L_1: L_1 = k, M_1 = 3, N_1 = -1 \))
- Line 2: \( 2x + y + 5 = 0 \) (Let \( L_2: L_2 = 2, M_2 = 1, N_2 = 5 \))

2. **Identify the circle's equation:**
The circle is given by:
\[ x^2 + y^2 - 2x - 4y - 4 = 0 \]
We can rewrite it in the standard form by completing the square.

3. **Complete the square:**
- For \( x \): \( x^2 - 2x \) can be rewritten as \( (x-1)^2 - 1 \)
- For \( y \): \( y^2 - 4y \) can be rewritten as \( (y-2)^2 - 4 \)

Thus, the equation becomes:
\[ (x-1)^2 + (y-2)^2 - 1 - 4 - 4 = 0 \]
Simplifying gives:
\[ (x-1)^2 + (y-2)^2 = 9 \]
This shows that the center of the circle is \( (1, 2) \) and the radius \( r = 3 \).

4. **Use the condition for conjugate lines:**
The condition for two lines \( L_1 \) and \( L_2 \) to be conjugate with respect to a circle is given by:
\[ r^2 (L_1 L_2) + (M_1 M_2) = (L_1 g + M_1 f - N_1)(L_2 g + M_2 f - N_2) \]
Where \( g \) and \( f \) are derived from the circle's equation.

5. **Identify \( g \) and \( f \):**
From the circle's equation:
\[ g = -1, \quad f = -2 \]

6. **Substituting values into the conjugate condition:**
- \( r^2 = 9 \)
- \( L_1 = k, M_1 = 3, N_1 = -1 \)
- \( L_2 = 2, M_2 = 1, N_2 = 5 \)

Now substituting these values:
\[ 9(k \cdot 2 + 3 \cdot 1) = (k(-1) + 3(-2) + 1)(2(-1) + 1(-2) - 5) \]

7. **Calculate each side:**
- Left side:
\[ 9(2k + 3) \]
- Right side:
\[ (-k - 6 + 1)(-2 - 2 - 5) = (-k - 5)(-9) = 9(k + 5) \]

8. **Set the equations equal:**
\[ 9(2k + 3) = 9(k + 5) \]
Dividing both sides by 9:
\[ 2k + 3 = k + 5 \]

9. **Solve for \( k \):**
\[ 2k - k = 5 - 3 \]
\[ k = 2 \]

### Final Answer:
The value of \( k \) is \( 2 \).