If the angle made by the lines represented by the equation ax^2 + 2hxy + by^2 = 0 with X-axis are and , then ( + ) is

Step-by-step solution

Let the slopes of the lines represented by ax^2 + 2hxy + by^2 = 0 be m₁ and m₂ . Since the lines make angles and with the X-axis, we have m₁ = and m₂ = . From the given homogeneous equation, dividing by x^2 gives b ( y x )^2 + 2h ( y x ) + a = 0 . The roots of this quadratic equation in m = y x are m₁ and m₂ . Sum of the roots: m₁ + m₂ = + = - 2h b Product of the roots: m₁ m₂ = = a b Using the trigonometric identity for the tangent of a sum: ( + ) = + 1 - Substituting the values of the sum and product of the slopes