Lines `mx + 3y = -7 ` and `5x - ny = 3 ` are perpendicular to each other. Find the relation connecting m and n. 

Answer

To find the relation connecting \( m \) and \( n \) for the lines \( mx + 3y = -7 \) and \( 5x - ny = 3 \) which are perpendicular to each other, we can follow these steps:

### Step 1: Identify the slopes of the lines
The general form of a line is given by \( Ax + By + C = 0 \). The slope \( m \) of the line can be calculated as \( m = -\frac{A}{B} \).

For the first line \( mx + 3y = -7 \):
- Rearranging gives \( 3y = -mx - 7 \) or \( y = -\frac{m}{3}x - \frac{7}{3} \).
- Thus, the slope \( m_1 = -\frac{m}{3} \).

For the second line \( 5x - ny = 3 \):
- Rearranging gives \( -ny = -5x + 3 \) or \( y = \frac{5}{n}x - \frac{3}{n} \).
- Thus, the slope \( m_2 = \frac{5}{n} \).

### Step 2: Use the condition for perpendicular lines
Two lines are perpendicular if the product of their slopes is equal to -1:
\[ m_1 \cdot m_2 = -1 \]

Substituting the slopes we found:
\[ \left(-\frac{m}{3}\right) \cdot \left(\frac{5}{n}\right) = -1 \]

### Step 3: Simplify the equation
This simplifies to:
\[ -\frac{5m}{3n} = -1 \]
Multiplying both sides by -1 gives:
\[ \frac{5m}{3n} = 1 \]

### Step 4: Cross-multiply to find the relation
Cross-multiplying gives:
\[ 5m = 3n \]

### Step 5: Rearranging the relation
We can express the relation as:
\[ \frac{m}{n} = \frac{3}{5} \]

### Final Relation
Thus, the relation connecting \( m \) and \( n \) is:
\[ 5m = 3n \]