The lines \( x=py+3,\ z=2y+2 \) and \( x=p'y+2,\ z=2y+3 \) are perpendicular if \( |pp'| = \)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 5
With \( y=t \) as the parameter, the first line is \( x=pt+3,\ y=t,\ z=2t+2 \), so its direction ratios are \( (p,\,1,\,2) \).
The second line is \( x=p't+2,\ y=t,\ z=2t+3 \), with direction ratios \( (p',\,1,\,2) \).
The lines are perpendicular when the dot product of their direction ratios is zero:
\( \displaystyle pp'+1+4=0 \)
\( \displaystyle pp'=-5 \)
Hence \( |pp'|=5 \).