A line L₁ passes through the point P(2,3) and has a slope of 1 2 . Another line L₂ also passes through P and intersects the x-axis at a point A . If the distance of A from the origin is 5 , then the sum of all possible values of , where is the acute angle between L₁ and L₂ , is equal to

Step-by-step solution

Since the point A lies on the x-axis and its distance from the origin is 5 , the coordinates of A can be either (5,0) or (-5,0) . Case 1: A is (5,0) . The slope of line L₂ passing through P(2,3) and A(5,0) is m₂ = 0 - 3 5 - 2 = -1 . The slope of L₁ is given as m₁ = 1 2 . The acute angle ₁ between L₁ and L₂ is given by: ₁ = | m₁ - m₂ 1 + m₁ m₂ | = | 1 2 - (-1) 1 + ( 1 2 )(-1) | = | 3 2 1 2 | = 3 . Case 2: A is (-5,0) . The slope of line L₂ passing through P(2,3) and A(-5,0) is m₂' = 0 - 3 -5 - 2 = 3 7 . The acute an