The lines $$x = ay + b,\,z = cy + d$$     and $$x = a'y + b',\,z = c'y + d'$$      will be perpendicular if and only if :

Solution :
For the first line, $$x - b = ay,\,z - d = cy\,\,\,\,\, \Rightarrow \frac{{x - b}}{a} = \frac{y}{1} = \frac{{z - d}}{c}.$$
For the second line, $$x - b' = a'y,\,z - d' = c'y\,\,\, \Rightarrow \frac{{x - b'}}{{a'}} = \frac{y}{1} = \frac{{z - d'}}{{c'}}.$$
These lines are perpendicular $$ \Rightarrow \,a.a' + 1.1 + c.c' = 0.$$