Two lines whose direction cosines are given by a l+b m+c n=0 and f m n+g n l+h l m=0 are perpendicular to each other if .........

Step-by-step solution

Let the direction cosines of lines are l₁, m₁, n₁ and l₂, m₂, n₂ Since, a l+b m+c n=0 and, f m n+g n l+h m=0 So, on eliminating ' l ', we get gathered f m n+(g n+h m) (- b m+c n a )=0 a f m n=b h m^2+c g n^2+(b g+c h) m n=0 b h ( m n )^2+(b g+c h-a f) ( m n )+c g=0 having roots m₁ n₁ and m₂ n₂ , so product of roots = m₁ m₂ n₁ n₂ = c g b h gathered Similarly on eliminating ' m ', we get gathered g n l+(h l+f n) (- a l+c n b )=0 b g n l=a h l^2+c f n^2+(c h+a f) l n=0 a h ( l n )^2+(c h+a f-b g) ( l n )+c f=0 , havin