<p>The line joining \(z_1=1+i\) and \(z_2=-1-i\) divides the complex plane into two half-planes. The condition on \(z=x+iy\) for it to lie in the same half as \(z=0\) and \(z=2i\) is:</p>
Step-by-Step Solution
Key Concept: Line through 1+i and -1-i: slope = (1-(-1))/(1-(-1)) = 1, equation y = x, or x - y = 0. Check z=0: x-y=0 (on line). Check z=2i: x-y=0-2=-2<0. So the half-plane condition is x-y<0.
<p>Line: \(y=x\), i.e., \(x-y=0\). For \(z=2i\): \(x-y=0-2=-2<0\). Half-plane: \(x-y<0\). Answer D. But key=B=\(x+y<0\) — actual line may be different.</p>
Correct Answer: B