If the set Im ( z + z + z z z + c ) : z C , Im (z) = 3 is equal to the interval [ 8 3 , 3 ) , where c is a real constant, then the value of c is

Step-by-step solution

Let z = x + 3i . The given expression is z + z + z z z + c . Substitute z = x + 3i : Numerator = (x + 3i) + (x - 3i) + (x^2 + 9) = x^2 + 2x + 9 Denominator = (x - 3i) + c = (x + c) - 3i The expression becomes x^2 + 2x + 9 (x + c) - 3i . To find the imaginary part, multiply the numerator and the denominator by the conjugate of the denominator, (x + c) + 3i : Im ( (x^2 + 2x + 9)((x + c) + 3i) (x + c)^2 + 9 ) = 3(x^2 + 2x + 9) (x + c)^2 + 9 Let y = 3(x^2 + 2x + 9) (x + c)^2 + 9 . Rearranging into a quadratic equation