If 4 ab =3 ~h ^2 , then the ratio of slopes of the lines represented by a x^2+2 h x y+b y^2=0 is

Step-by-step solution

We have a x^2+2 h x y+b^2=0 and let m₁ and m₂ be the slopes of lines. Now m ₁+ m ₂= -2 ~h ~b and m ₁ ~m ₂= a b aligned & ( m ₁- m ₂ )^2= ( m ₁+ m ₂ )^2-4 ~m ₁ ~m ₂ & = ( -2 ~h ~b )-4 ( a b )= 4 ~h ^2 ~b ^2 - 4 a b = 4 ~h ^2-4 ab b ^2 = 4 ~h ^2-3 ~h ^2 ~b ^2 aligned ...[From data given] gathered = h ^2 ~b ^2 m ₁- m ₂= h b gathered Thus we have m ₁+ m ₂= -2 ~h ~b and m ₁- m ₂= h b Solving, we get m ₁= - h 2 ~b and m ₂= -3 ~h 2 ~b m ₁: m ₂=1: 3