The larger of two angles made with the X -axis of a straight line drawn through ( 1 , 2 ) so that it intersects the line x + y = 4 at a point distant 6 / 3 from the point ( 1 , 2 ) is

Step-by-step solution

Let A = ( 1 , 2 ) equation of line is x - 1 cos θ = y - 2 sin θ = r = 6 3 ∴   x = 6 3 cos θ + 1 , y = 6 3 sin θ + 2 This point lies on line x + y = 4 ∴   6 3 cos θ + 6 3 sin θ = 1 ⇒   sin θ + cos θ = 3 2 ⇒   sin π 4 + θ = 3 2 ⇒   π 4 + θ = 60 ° or 120 ° ⇒   θ = 15 ° or 75 ° ∴ Larger angle = 75 °