Question

If the lines x = ay + b, z = cy + d and x = a'z + b', y = c'z + d' are perpendicular, then:
(A) cc' + a + a' = 0 (B) aa' + c + c' = 0
(C) ab' + bc' + 1 = 0 (D) bb' + cc' + 1 = 0

Answer 1

If the lines x = ay + b, z = cy + d and x = a'z + b', y = c'z + d' are perpendicular, then,  aa' + c' + c = 0

Consider first line,

• (x -b) / a = y -- (1a)
• ( z - d)/ c = y--(2a)
• Therefore the line equation is (x-b)/a  = (y-0)/1 = (z -d)/c
• Therefore the parallel vector along the line is V1= ai + 1j + ck --(3a)

Consider second line,

• (x - b')/a' = z --(1b)
• (y - d')/c' = z --(2b)
• Therefore line equation is (x-b')a' = (y - d')/c' = (z - 0)/1
• Therefore parallel vector along this line is V2 = a'i + c'j + 1k

Given  the lines are perpendicular.

• This implies, dot product of V1 and V2 is zero.

• V1.V2 = 0
• (ai + j + ck).(a'i + c'j + k) = 0==>
• aa' + c' + c = 0

Answer is option B .