The two equations are `4 x - 4 ky - 6 = 0 and x = 6 y - 12 = 0 `. Find the value of k for which the pair of two linear equations represents the parallel lines

Options

A. 2
B. 6
C. 8
D. 10

Answer

To find the value of \( k \) for which the two equations represent parallel lines, we need to analyze the given equations:

1. The first equation is:
\[ 4x - 4ky - 6 = 0 \]

2. The second equation is:
\[ x - 6y - 12 = 0 \]

### Step 1: Rewrite both equations in standard form
The standard form of a linear equation is \( Ax + By + C = 0 \).

For the first equation:
\[ 4x - 4ky - 6 = 0 \]
Here, \( A_1 = 4 \), \( B_1 = -4k \), and \( C_1 = -6 \).

For the second equation:
\[ x - 6y - 12 = 0 \]
Here, \( A_2 = 1 \), \( B_2 = -6 \), and \( C_2 = -12 \).

### Step 2: Use the condition for parallel lines
For two lines to be parallel, the ratios of the coefficients of \( x \) and \( y \) must be equal:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} \]

Substituting the values we found:
\[ \frac{4}{1} = \frac{-4k}{-6} \]

### Step 3: Simplify the equation
This simplifies to:
\[ 4 = \frac{4k}{6} \]

### Step 4: Cross-multiply to solve for \( k \)
Cross-multiplying gives:
\[ 4 \cdot 6 = 4k \]
\[ 24 = 4k \]

### Step 5: Solve for \( k \)
Now, divide both sides by 4:
\[ k = \frac{24}{4} = 6 \]

Thus, the value of \( k \) for which the two equations represent parallel lines is:
\[ \boxed{6} \]