Let P = z ∈ ℂ : z + 2 − 3 i ≤ 1 and Q = z ∈ ℂ : z 1 + i + z ¯ 1 − i ≤ − 8 . Let in P ∩ Q , z − 3 + 2 i be maximum and minimum at z 1 and z 2 respectively. If z 1 2 + 2 z 2 = α + β 2 , where α , β are integers, then α + β equals __________

Step-by-step solution

Given: z + 2 - 3 i ≤ 1 Putting, z = x + i y . ⇒ x + 2 2 + y - 3 2 ≤ 1 . . . i For the circle represented in equation i , centre is - 2 , 3 and radius, r = 1 . It is given that, z 1 + i + z 1 - i ≤ - 8 ⇒ x + i y 1 + i + x - i y 1 - i ≤ - 8 ⇒ x + i x - y + i y + x - i x - i y - y ≤ - 8 ⇒ 2 x - 2 y ≤ - 8 ⇒ x - y + 4 = 0 So, the line passing through 3 , - 2 and perpendicular to x - y + 4 = 0 is given by, x + y - 1 = 0 . So, the distance of line x - y + 4 = 0 from the centre - 2 , 3 is given by, d = - 2 - 3 + 4 2 ⇒ d =