JEE MAIN - Mathematics (2003 - No. 28)
The two lines $$x=ay+b,z=cy+d$$ and $$x = a'y + b',z = c'y + d'$$ will be perpendicular, if and only if :
$$aa' + cc' + 1 = 0$$
$$aa' + bb'cc' + 1 = 0$$
$$aa' + bb'cc' = 0$$
$$\left( {a + a'} \right)\left( {b + b'} \right) + \left( {c + c'} \right) = 0$$
Explanation
$${{x - b} \over a} = {y \over 1} = {{z - d} \over c};$$
$${{x - b'} \over {a'}}$$ $$ = {y \over 1} = {{z - d'} \over c'}$$
For perpenedicularity of lines $$aa' + 1 + cc' = 0$$
$${{x - b'} \over {a'}}$$ $$ = {y \over 1} = {{z - d'} \over c'}$$
For perpenedicularity of lines $$aa' + 1 + cc' = 0$$
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