Question

The lines \[ p(p^2 + 1)x - y + q = 0 \quad {and} \quad (p^2 + 1)^2 x + (p^2 + 1)y + 2q = 0 \] are perpendicular to a common line for:

• A. Exactly one value of \( p \)
• B. Exactly two values of \( p \)
• C. More than two values of \( p \)
• D. No value of \( p \)

Hint:

If two lines are perpendicular to the same line, their slopes satisfy \( m_1 m_2 = -1 \).

Answer 1

The Correct Option is A

Step 1: Finding slopes of given lines. 
Rewriting equations in slope-intercept form: 
- The first line has slope \( m_1 = p(p^2 + 1) \). 
- The second line has slope \( m_2 = -(p^2 + 1) / (p^2 + 1)^2 \). 
Step 2: Condition for perpendicularity. 
For the lines to be perpendicular to a common line, \[ m_1 m_2 = -1 \] Solving for \( p \), we find only one valid solution.