If the lines y + px = 1 and y-qx = 2 are perpendicular, then which one of the following is correct?

Options

A. `pq+1=0`
B. `p+q+1=0`
C. `pq-1=0`
D. `p-q+1=0`

Answer

To solve the problem of determining the relationship between \( p \) and \( q \) for the lines \( y + px = 1 \) and \( y - qx = 2 \) to be perpendicular, we can follow these steps:

### Step 1: Rewrite the equations in slope-intercept form
We start with the two equations:
1. \( y + px = 1 \)
2. \( y - qx = 2 \)

We need to express both equations in the form \( y = mx + c \), where \( m \) is the slope.

For the first equation:
\[ y + px = 1 \implies y = 1 - px \implies y = -px + 1 \]
Here, the slope \( m_1 = -p \).

For the second equation:
\[ y - qx = 2 \implies y = 2 + qx \implies y = qx + 2 \]
Here, the slope \( m_2 = q \).

### Step 2: Use the condition for perpendicular lines
Two lines are perpendicular if the product of their slopes is \(-1\):
\[ m_1 \cdot m_2 = -1 \]
Substituting the slopes we found:
\[ (-p) \cdot q = -1 \]

### Step 3: Simplify the equation
This simplifies to:
\[ -pq = -1 \]
Multiplying both sides by \(-1\) gives:
\[ pq = 1 \]

### Step 4: Conclusion
Thus, the relationship between \( p \) and \( q \) is:
\[ pq - 1 = 0 \]

### Final Answer
The correct option is \( pq - 1 = 0 \).

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