Find the relation between `a` and `b` if the lines `3x-by+5=0` and `ax+y=2` parallel.

Answer

To find the relation between \( a \) and \( b \) for the lines \( 3x - by + 5 = 0 \) and \( ax + y = 2 \) to be parallel, we follow these steps:

### Step 1: Write the equations in slope-intercept form
We need to express both equations in the form \( y = mx + c \), where \( m \) is the slope.

1. For the first line \( 3x - by + 5 = 0 \):
\[ -by = -3x - 5 \implies y = \frac{3}{b}x + \frac{5}{b} \]
Here, the slope \( m_1 = \frac{3}{b} \).

2. For the second line \( ax + y = 2 \):
\[ y = -ax + 2 \]
Here, the slope \( m_2 = -a \).

### Step 2: Set the slopes equal
Since the lines are parallel, their slopes must be equal:
\[ m_1 = m_2 \]
This gives us:
\[ \frac{3}{b} = -a \]

### Step 3: Rearrange to find the relationship
Now, we can rearrange the equation to find the relationship between \( a \) and \( b \):
\[ -a = \frac{3}{b} \]
Multiplying both sides by \( b \) (assuming \( b \neq 0 \)):
\[ -ab = 3 \]
Thus, we can express this as:
\[ ab = -3 \]

### Final Result
The relation between \( a \) and \( b \) is:
\[ ab = -3 \]