Prove that the lines `x = py + q; z = ry + s and x = p'y + q'; z = r'y + s'` are perpendicular If `p p' + q q' + 1 = 0`

Explanation

We have `x=py+qimpliesy=(x-q)/P`……………..i
and `z=ry+simpliesy=(z-s)/r`………………ii
`implies(x-q)/p=y/1=(z-s)/r` [using eqs i and ii]…………..iii
Similarly, `(x-q')/(p')=y/1=(z-s')/(r')`…………………iv
From Eqs. (iii) and (iv)
`a_(1)=p,b_(1)=1,c_(1)=r` ltbr. and `a_(2)=p',b_(2)=1,c_(2)=r'`
If these given lines are perpendicular to each other, then
`a_(1)a_(2)+b_(1)b_(2)+c_(1)c_(2)=0`
`impliespp'+1+n'=0`
which is the required condition.