Question

• A. $(2-1/\sqrt{2})(1-i)$
• B. $(2-1/\sqrt{2})(1+i)$
• C. $(2+1/\sqrt{2})(1-i)$
• D. $(2+1/\sqrt{2})(1+i)$

Answer 1

The correct option is A $(2-1/\sqrt{2})(1-i)$

We have,
$|z-2+2i|=1$
$\Rightarrow |z-(2-2i)|=1$
Hence, $z$ lies on a circle having center at $(2,-2)$ and radius $1$. It is evident from the figure that the required complex number $z$ is given by the point $P$.
We find that $OP$ makes an angle $\pi /4$ with $OX$ and
$PO=OC-CP=\sqrt{2^2+2^2}-1=2\sqrt{2}-1$
So, coordinates of $P$ are $[2\sqrt{2}-1)cos\left(\pi /4\right)$,
$-(2\sqrt{2}-1)sin\left(\pi /4\right)],i.e.,(2-1/\sqrt{2}),-(2-1/\sqrt{2}\left)\right)$. Hence,
$z=(2-\frac{1}{\sqrt{3}})+\{-(2-\frac{1}{\sqrt{2}})\}i=(2-\frac{1}{\sqrt{2}})(1-i)$