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If lines x=ay+b, z= cy +d " and " x=a...

If lines `x=ay+b, z= cy +d " and " x=a' z+b`
`y=c' z+ d' ` are perpendicular then

A

`ab+bc+1=0`

B

`bb+cc'+1=0`

C

`aa'+c+c'=0`

D

`cc'+a+a'=0`

Text Solution

Verified by Experts

The correct Answer is:
C
Let `1^(st)` line is x = ay +b, z =cy +d.
`rArr (x-b)/(a) y , (z-d)/( c) =y rArr (x-b)/(a) =y= (z-d)/(c )`
the direction vector of this line is `b_(1)= ahat(i) +hat(j) + chat(k).`
Let `2^(nd)` line is x= a'z +b', y =c' z+d'
`rArr (x-b)/(a)=z, (y-d)/(c ) =z rArr (x-b)/(a )' =(y-d)/( c) =z`
The direction vector of this line is `b_(2)= a'hat(i) +c'hat(j) + hat(k)`
`:. ` The two lines are perpendicular therefore `b_(1). b_(2) =0`
`rArr(ahat(i) +hat(j) +chat(k)) . (a'hat(i)+chat(j) +hat(k)) =0`
`rArr aa' +c' +c=0 " " rArr aa' +c+ c=0`
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