If the slope of one of the two lines given by x^2 a + 2xy h + y^2 b = 0 is twice that of the other, then h^2 : ab = ...

Step-by-step solution

The given equation is x^2 a + 2xy h + y^2 b = 0 Dividing by x^2 , we get 1 b ( y x )^2 + 2 h ( y x ) + 1 a = 0 Let m = y x represent the slope of the lines. The equation becomes m^2 b + 2m h + 1 a = 0 Let the slopes of the two lines be m₁ and m₂ . From the quadratic equation in m , the sum and product of the roots are: m₁ + m₂ = - 2b h m₁ m₂ = b a Given that the slope of one line is twice that of the other, m₂ = 2m₁ Substituting m₂ in the sum of the roots: m₁ + 2m₁ = - 2b h 3m₁ = - 2b h m₁ = - 2b 3h Substituting m₂