Let z₁ and z₂ be the roots of the equation z^4 + 16 = 0 that lie in the upper half of the complex plane. Let L be the line passing through z₁ and z₂ . If L divides the circular region |z| 2 into two parts of areas and ( > ), then the value of - is :

Step-by-step solution

The roots of z^4 + 16 = 0 are given by z = (-16)^ 1/4 = (16 e^ i )^ 1/4 . z = 2 e^ i( + 2k )/4 for k = 0, 1, 2, 3 . For k = 0 , z₁ = 2 e^ i /4 = 2 + i 2 . For k = 1 , z₂ = 2 e^ i3 /4 = - 2 + i 2 . For k = 2 , z₃ = 2 e^ i5 /4 = - 2 - i 2 . For k = 3 , z₄ = 2 e^ i7 /4 = 2 - i 2 . The roots in the upper half plane (where the imaginary part is positive) are z₁ and z₂ . The line L passing through z₁ and z₂ is horizontal, with equation y = 2 . The region |z| 2 represents a circle centered at the origin (0,0) with radius