Two of the lines represented by the equation a y^4+b x y^3+c x^2 y^2+d x^3 y+e x^4=0 will be perpendicular, then

Step-by-step solution

Let a y^4+b x y^3+c x^2 y^2+d x^3 y+e x^4= (a x^2+p x y-a y^2 ) (x^2+q x y+y^2 ) Comparing the coefficients of similar terms, b = aq − p, c = − pq, d = aq + p and e = − a b + d = 2aq and e − a = − 2a ad + be = 2ap and a + c + e = − pq aligned & (b+d)(a d+b e)=(2 a q) 2 a p=4 a^2 p q & Also, -(e-a)^2(a+c+e)=-(-2 a)^2(-p q)=4 a^2 p q & (b+d)(a d+b e)=-(e-a)^2(a+c+e) & (b+d)(a d+b e)+(e-a)^2(a+c+e)=0 aligned