If the line segment joining the points A(3, 3) and B(k, -2) subtends an angle of 4 at the point P(2, 1) , then the sum of the squares of all possible values of k is :

Step-by-step solution

The slope of the line PA is m₁ = 3 - 1 3 - 2 = 2 . The slope of the line PB is m₂ = -2 - 1 k - 2 = -3 k - 2 . The angle between PA and PB is 4 . Using the formula for the angle between two lines: ( 4 ) = | m₁ - m₂ 1 + m₁ m₂ | 1 = | 2 - ( -3 k - 2 ) 1 + 2 ( -3 k - 2 ) | 1 = | 2(k - 2) + 3 k - 2 - 6 | = | 2k - 1 k - 8 | This gives two cases: Case 1: 2k - 1 k - 8 = 1 2k - 1 = k - 8 k = -7 Case 2: 2k - 1 k - 8 = -1 2k - 1 = -k + 8 3k = 9 k = 3 The possible values of k are -7 and 3 . The sum of their squares is (-7)^2 +