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Question

Prove that the lines x=py+q,z=ry+s and x=p'y+q',z=r'y+s' are perpendicular, if pp'+n'+1=0.


Solution

We have, x=py+qy=xqP ...(i)
and z=ry+sy=zsr ...(ii)
xqp=y1=zsr [using Eqs.(i) and (ii)] ...(iii)
Similarly, xqp=y1=zsr ...(iv)
From Eqs. (iii) and (iv), a1=p,b1=1,c1=r and a2=p,b2=1,c2=r
If these given lines are perpendicular to each other, then 
a1a2+b1b2+c1c2=0pp+1+n=0
which is the required condition. 

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