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Question

The lines x=ay+b,z=cy+d and x=ay+b,z=cy+d will be perpendicular if and only if

A
aa+bb+cc=0
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B
(a+a)(b+b)+c+c=0
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C
aa+cc+1=0
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D
aa+bb+cc+1=0
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Solution

The correct option is C aa+cc+1=0
The two given lines are:-
L1x=ay+b,z=cy+d
y=xba,y=zdc
we can write L1 as
L1xba=zdc=y
Similarly,
L2x=ay+b,z=cy+d
y=xba,y=zdc
L2 can be written as:-
L2xba=zdc=y
Direction ratios of line L1(a,c,1)
and
Direction ratios of line L2(a,c,1)
We know that two lines are perpendicular iff, product of their direction ratios are 0.
Therefore,
a×a+c×c+1×1=0
aa+cc+1=0
Hence, C is the correct answer.

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