If the lines \( x=ay+b \), \( z=cy+d \) and \( x=py+q \), \( z=ry+s \) are perpendicular, then \( ap+cr \) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → −1
The direction ratios are read from the symmetric form of each line. For the first line, \( y=\frac{x-b}{a} \) and \( y=\frac{z-d}{c} \):
\( \displaystyle \frac{x-b}{a}=\frac{y-0}{1}=\frac{z-d}{c} \)
Its direction ratios are a, 1, c. In the same way the second line has direction ratios p, 1, r.
The lines are perpendicular when \( a_1a_2+b_1b_2+c_1c_2=0 \):
\( \displaystyle ap+(1)(1)+cr=0 \)
\( \displaystyle ap+cr=-1 \)
Why the other options are incorrect
Option (1) 2 — Incorrect. With ap + cr = 2, the sum \( ap+1+cr \) is 3, not 0.
Option (2) 0 — Incorrect. 0 leaves out the product of the y-components, which is (1)(1) = 1.
Option (4) 1 — Incorrect. 1 has the wrong sign; moving 1 to the other side gives −1.