Question

The locus of complex number $z$ satisfying $\arg [z-(1+i\left)\right]=\{\begin{array}{cc}\frac{3\pi }{4}& ;|z|\le |z-2|\\ -\frac{\pi }{4}& ;|z|>|z-2|\end{array}$ is

• A. a straight line passing through $(2,0)$.
• B. a straight line passing through $(2,0),(1,1)$.
• C. is a line segment.
• D. is a set of two rays.

Answer 1

The correct option is D is a set of two rays.

Given equation is $\arg [z-(1+i\left)\right]=\{\begin{array}{cc}\frac{3\pi }{4}& ;|z|\le |z-2|\\ -\frac{\pi }{4}& ;|z|>|z-2|\end{array}$
Let $z=x+iy$
When $|z|\le |z-2|$
$\Rightarrow x^2+y^2\le (x-2{)}^2+y^2\Rightarrow 0\le -4x+4\Rightarrow x\le 1$
When $|z|>|z-2|$
$\Rightarrow x>1$

Now,
$\arg [z-(1+i\left)\right]=\{\begin{array}{cc}\frac{3\pi }{4}& ;Re\left(z\right)\le 1\\ -\frac{\pi }{4}& ;Re\left(z\right)>1\end{array}$


Therefore, the locus is a set of two rays.