JEE MAIN - Mathematics (2006 - No. 14)
The two lines $$x=ay+b, z=cy+d;$$ and $$x=a'y+b' ,$$ $$z=c'y+d'$$ are perpendicular to each other if :
$$aa'+cc'=-1$$
$$aa'+cc'=1$$
$${a \over {a'}} + {c \over {c'}} = - 1$$
$${a \over {a'}} + {c \over {c'}} = 1$$
Обяснение
Equation of lines
$${{x - b} \over a} = {y \over 1} = {{z - d} \over c}$$
$${{x - b'} \over {a'}} = {y \over 1} = {{z - d'} \over {c'}}$$
Line are perpendicular
$$ \Rightarrow aa' + 1 + cc' = 0$$
$${{x - b} \over a} = {y \over 1} = {{z - d} \over c}$$
$${{x - b'} \over {a'}} = {y \over 1} = {{z - d'} \over {c'}}$$
Line are perpendicular
$$ \Rightarrow aa' + 1 + cc' = 0$$