Prove that the lines $x = py + q,z = ry + s$ and $x$
$= {p^\prime }y + {q^\prime },z = {r^\prime }y + {s^\prime }$ are perpendicular, if $p{p^\prime } + r{r^\prime } + 1 = 0$.

It is given that, $x = py + q \Rightarrow y = \frac{{x - q}}{P}$

……(i)
and $z = ry + s \Rightarrow y = \frac{{z - s}}{r}$

…..(ii)
$\Rightarrow$ $\frac{{x - q}}{p} = \frac{y}{1} = \frac{{z - s}}{r}$

[using Eqs. (i) and(ii)] ….(iii)

Similarly, $\frac{{x - {q^\prime }}}{{{p^\prime }}} = \frac{y}{1} = \frac{{z - {s^\prime }}}{{{r^\prime }}}$

…….(iv)
From Eqs. (iii) and (iv),
${a_1} = p,{b_1} = 1,{c_1} = r$

and ${a_2} = {p^\prime },{b_2} = 1,{c_2} = {r^\prime }$

If these given lines are perpendicular to each other, then

${a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0$

$\Rightarrow$ $p{p^\prime } + 1 + r{r^\prime } = 0$

which is the required condition.