If the set Re z - z ¯ + z z ¯ 2 - 3 z + 5 z ¯ : z ∈ ℂ , Re z = 3 is equal to the interval ( α , β ] , then 24 β - α is equal to

Step-by-step solution

Given that the set Re z - z ¯ + z z ¯ 2 - 3 z + 5 z ¯ : z ∈ ℂ ,   Re z = 3 is equal to the interval ( α , β ] . Let z = x + i y = Re x + i y - x - i y + x 2 + y 2 2 - 3 x + i y + 5 x - i y = Re x 2 + y 2 + i 2 y 2 + 2 x - 8 i y = Re x 2 + y 2 + 2 y i 2 1 + x + 8 i y 2 1 + x 2 + 8 y 2 = 2 x 2 + y 2 1 + x - 16 y 2 4 1 + x 2 + 8 y 2 Now using, R e z = 3 ⇒ x = 3 we get, = 8 9 + y 2 - 16 y 2 64 + 64 y 2 ⇒ f y = 1 8 9 - y 2 1 + y 2 Let t = 1 8 9 - y 2 1 + y 2 ⇒