The slopes of the lines represented by 6 x^2+2 ~h x y + y ^2=0 are in the ratio 2: 3 , then h =

Step-by-step solution

The equation 6x^2 + 2hxy + y^2 = 0 represents a pair of lines through the origin. Comparing with the standard form ax^2 + 2h'xy + by^2 = 0 gives a = 6 , h' = h , and b = 1 . The slopes m₁ and m₂ satisfy m₁ + m₂ = -2h and m₁ m₂ = 6 . Given the ratio m₁ : m₂ = 2 : 3 , let m₁ = 2 3 m₂ . Substituting into the product gives 2 3 m₂^2 = 6 , so m₂^2 = 9 and m₂ = 3 . For m₂ = 3 , m₁ = 2 and m₁ + m₂ = -2h yields 5 = -2h , so h = - 5 2 . For m₂ = -3 , m₁ = -2 and m₁ + m₂ = -2h yields -5 = -2h , so h = 5 2 . The values of h ar