If two lines represented by x^2 - (1 + 3 )xy + 3 y^2 = 0 make angles and with the X-axis, then ( + ) is...

Step-by-step solution

The given equation of the pair of straight lines is x^2 - (1 + 3 )xy + 3 y^2 = 0 . Dividing the entire equation by x^2 , we get: 3 ( y x )^2 - (1 + 3 ) ( y x ) + 1 = 0 Let m = y x . The slopes of the two lines, m₁ = and m₂ = , are the roots of the quadratic equation: 3 m^2 - (1 + 3 )m + 1 = 0 From the properties of quadratic equations, the sum and product of the roots are: + = 1 + 3 3 = 1 3 Using the trigonometric identity for the tangent of a sum: ( + ) = + 1 - Substituting the values, we get: ( + ) = 1 + 3 3 1 -