If a d 0 and two of the lines represented by a x^3+3 b x^2 y+3 cxy ^2+ dy ^3=0 are perpendicular, then

Step-by-step solution

a x^3+3 b x^2 y+3 c x y^2+d y^3=0...(i) is a homogeneous equation of third degree is x & y . Let the slopes of the lines be m₁, m₂, m₃ . Then, m₁, m₂ and m₃ are the roots of dm ^3+3 ~cm ^2+3 bm +a=0 ...(ii) Product of roots =m₁ m₂ m₃=- a d ...(iii) Since, the two lines represented by (i) are perpendicular angles. Let the lines with slopes m₁, m₂ be perpendicular m₁ m₂=-1 From equation (iii), (-1) m₃=- a d m₃= a d ...(iv) But m ₃ is root of equation (ii): aligned & d ( a d )^3+3 c ( a d )^2+3 b ( a d )+a=0 & a^3+3 a