Let z be those complex numbers which satisfy z + 5 ≤ 4 and z 1 + i + z ¯ 1 - i ⩾ - 10 , i = - 1 . If the maximum value of | z + 1 | 2 is α + β 2 , then the value of α + β is

Step-by-step solution

Let z = x + i y Given | z + 5 | ≤ 4 x + 5 2 + y 2 ≤ 16     . . . 1 z 1 + i + z ¯ 1 - i ≥ - 10 z + z ¯ + i z - z ¯ ≥ - 10 x - y + 5 ≥ 0       . . . 2 | z + 1 | 2 = | z - - 1 | 2 Let P - 1 , 0 | z + 1 | Max . 2 = P B 2 (where B is in 3 rd quadrant) for point of intersection x + 5 2 + y 2 = 16 x - y + 5 = 0 y = ± 2 2 A 2 2 - 5 , 2 2   B - 2 2 - 5 , - 2 2 P B 2 = + 2 2 + 4 2 + 2 2 2 | z + 1 | 2 = 8 + 16 + 16 2 + 8 α + β 2 = 32 + 1