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

Step-by-step solution

The slope of the line 3 x+y-6=0 is m₁=-3 . Let m be the slope of one of the lines making an angle 6 with 3 x+y-6=0 . aligned & 6 = | m - m ₁ 1+ m ₁ | & 1 3 = | m -(-3) 1+ m (-3) | & 1 3 = | m +3 1-3 ~m | aligned Squaring on both sides, we get aligned & (1-3 m)^2=3(m+3)^2 & 6 m^2-24 m-26=0 & 3 m^2-12 m-13=0 aligned This is the auxiliary equation of two lines and their joint equation is obtained by putting m = y x . The joint equation of the lines is aligned & 3 ( y x )^2-12 ( y x )-13=0 & 13 x^2+12 x y-3 y^2=0 align