If Lines `2x+3y=18` and `4x+ky=39` are perpendicular line then find k?.

Answer

To find the value of \( k \) such that the lines \( 2x + 3y = 18 \) and \( 4x + ky = 39 \) are perpendicular, we can follow these steps:

### Step 1: Find the slope of the first line
The equation of the first line is given as:
\[ 2x + 3y = 18 \]
We can rearrange this equation to the slope-intercept form \( y = mx + c \).

1. Subtract \( 2x \) from both sides:
\[ 3y = -2x + 18 \]
2. Divide everything by \( 3 \):
\[ y = -\frac{2}{3}x + 6 \]
Thus, the slope \( m_1 \) of the first line is:
\[ m_1 = -\frac{2}{3} \]

### Step 2: Find the slope of the second line
The equation of the second line is:
\[ 4x + ky = 39 \]
Similarly, we rearrange this to find its slope.

1. Subtract \( 4x \) from both sides:
\[ ky = -4x + 39 \]
2. Divide everything by \( k \):
\[ y = -\frac{4}{k}x + \frac{39}{k} \]
Thus, the slope \( m_2 \) of the second line is:
\[ m_2 = -\frac{4}{k} \]

### Step 3: Use the condition for perpendicular lines
For two lines to be perpendicular, the product of their slopes must equal \(-1\):
\[ m_1 \cdot m_2 = -1 \]
Substituting the slopes we found:
\[ \left(-\frac{2}{3}\right) \cdot \left(-\frac{4}{k}\right) = -1 \]
This simplifies to:
\[ \frac{8}{3k} = -1 \]

### Step 4: Solve for \( k \)
To solve for \( k \), we cross-multiply:
\[ 8 = -3k \]
Now, divide both sides by \(-3\):
\[ k = -\frac{8}{3} \]

### Final Answer
The value of \( k \) is:
\[ \boxed{-\frac{8}{3}} \]