Let a = p + iq be a complex number where p > 0 and q > 0 . Define the set S = z C : Re (a z ) > Im ( a z) . Which of the following rays in the complex plane is strictly guaranteed to be entirely contained within the set S ?

Step-by-step solution

Let z = x + iy . We have a = p + iq and z = x - iy . Then a z = (p+iq)(x-iy) = (px+qy) + i(qx-py) . So, Re (a z ) = px+qy . Similarly, a z = (p-iq)(x+iy) = (px+qy) + i(py-qx) . So, Im ( a z) = py-qx . The given condition for set S is Re (a z ) > Im ( a z) , which gives: px+qy > py-qx (p+q)x + (q-p)y > 0 We are given that p > 0 and q > 0 , which implies p+q > 0 . Let us test the standard rays: For the positive real axis, y = 0 and x > 0 . The inequality becomes (p+q)x > 0 . Since p+q > 0 and x > 0 , this is always t