The joint equation of the pair of lines through the origin and making an angle of 4 with the line 3x + y - 6 = 0 is

Step-by-step solution

Slope of the given line 3x + y - 6 = 0 is m₁ = -3 . Let m be the slope of a line passing through the origin and making an angle of 4 with the given line. Using the angle formula, ( 4 ) = | m - m₁ 1 + m m₁ | 1 = | m - (-3) 1 + m(-3) | = | m + 3 1 - 3m | Squaring both sides, we get: (1 - 3m)^2 = (m + 3)^2 1 - 6m + 9m^2 = m^2 + 6m + 9 8m^2 - 12m - 8 = 0 Dividing by 4 , we get 2m^2 - 3m - 2 = 0 . Since the lines pass through the origin, their equations are of the form y = mx , which implies m = y x . Substituting m = y