Find the value of k for which the lines `kx - 5y + 4 = 0 ` and `5x – 2y + 5 = 0 ` are perpendicular to each other 

Answer

To find the value of \( k \) for which the lines \( kx - 5y + 4 = 0 \) and \( 5x - 2y + 5 = 0 \) are perpendicular, we can follow these steps:

### Step 1: Write the equations in slope-intercept form
We start with the first line:
\[ kx - 5y + 4 = 0 \]
Rearranging this equation gives:
\[ 5y = kx + 4 \]
Dividing by 5:
\[ y = \frac{k}{5}x + \frac{4}{5} \]
Thus, the slope \( m_1 \) of the first line is:
\[ m_1 = \frac{k}{5} \]

Now, for the second line:
\[ 5x - 2y + 5 = 0 \]
Rearranging this gives:
\[ 2y = 5x + 5 \]
Dividing by 2:
\[ y = \frac{5}{2}x + \frac{5}{2} \]
Thus, the slope \( m_2 \) of the second line is:
\[ m_2 = \frac{5}{2} \]

### Step 2: 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:
\[ \left(\frac{k}{5}\right) \cdot \left(\frac{5}{2}\right) = -1 \]

### Step 3: Solve for \( k \)
Now we simplify the equation:
\[ \frac{k \cdot 5}{5 \cdot 2} = -1 \]
The \( 5 \) in the numerator and denominator cancels out:
\[ \frac{k}{2} = -1 \]
Multiplying both sides by 2 gives:
\[ k = -2 \]

### Final Answer
Thus, the value of \( k \) for which the lines are perpendicular is:
\[ \boxed{-2} \]