If the center and radius of the circle z - 2 z - 3 = 2 are respectively α , β and γ , then 3 α + β + γ is equal to

Step-by-step solution

Given: z - 2 z - 3 = 2 Let z = x + i y , then we have x - 2 + i y x - 3 + i y = 2 ⇒ x - 2 2 + y 2 = 2 x - 3 2 + y 2 ⇒ x 2 + y 2 - 4 x + 4 = 4 x 2 + 4 y 2 - 24 x + 36 ⇒ 3 x 2 + 3 y 2 - 20 x + 32 = 0 ⇒ x 2 + y 2 - 20 3 x + 32 3 = 0 So, α , β ≡ 10 3 , 0 And, γ = 100 9 - 32 3 = 4 9 = 2 3 Therefore, 3 α + β + γ = 3 10 3 + 2 3 = 12