If the two lines given by
\( x=py+q,\ z=ry+s \) and
\( x=p'y+q',\ z=r'y+s' \)
are perpendicular, then which of the following is equal to -1?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( pp'+rr' \)
Taking y as the parameter, \( x=py+q \) and \( z=ry+s \) give
\( \displaystyle \frac{x-q}{p}=\frac{y}{1} \)
\( \displaystyle =\frac{z-s}{r} \)
So the direction ratios of the first line are \( (p,1,r) \), and in the same way those of the second line are \( (p',1,r') \).
The lines are perpendicular when the sum of the products of direction ratios is zero.
\( \displaystyle pp'+1\cdot1+rr'=0 \)
\( \displaystyle pp'+rr'=-1 \)