If 4ab = 3h^2 , then the ratio of the slopes of the lines represented by ax^2 + 2hxy + by^2 = 0 is...

Step-by-step solution

Let the slopes of the lines represented by ax^2 + 2hxy + by^2 = 0 be m₁ and m₂ . We know that: m₁ + m₂ = - 2h b m₁ m₂ = a b Squaring the sum of the slopes and dividing by their product, we get: (m₁ + m₂)^2 m₁ m₂ = (- 2h b )^2 a b = 4h^2 ab We are given that 4ab = 3h^2 , which can be rewritten as h^2 ab = 4 3 . Substituting this value into the equation: (m₁ + m₂)^2 m₁ m₂ = 4 ( 4 3 ) = 16 3 Let the ratio of the slopes be m₁ : m₂ = k : 1 , which means m₁ = k m₂ . Substituting this into the above relation: (k m₂ + m₂)^