Let S = z ∈ ℂ : z - 3 ≤ 1 and z 4 + 3 i + z ¯ 4 - 3 i ≤ 24 . If α + i β is the point in S which is closest to 4 i , then 25 α + β is equal to ______.

Step-by-step solution

Given, z - 3 ≤ 1 Which represent a circle of radius 1 and centred at 3 , 0 Now simplifying z 4 + 3 i + z ¯ 4 - 3 i ≤ 24 We get, x + i y 4 + 3 i + x - i y 4 - 3 i ≤ 24 ⇒ 4 x + 3 x i + 4 i y - 3 y + 4 x - 3 i x - 4 i y - 3 y ≤ 24 ⇒ 8 x - 6 y ≤ 24 ⇒ 4 x - 3 y ≤ 12 Now plotting the diagram we get, Minimum distance of 0 , 4 from circle = 3 2 + 4 2 - 1 = 4 will lie along line joining 0 , 4   &   3 , 0 ∴ equation line x 3 + y 4 = 1 ⇒ 4 x