Question

Locate the points representing the complex number for which ${\log }_{1/2}\frac{|z-1|+4}{|z-1|-2}>1$

• A. $|z-1|<2$
• B. $|z-1|>2$
• C. $|z-1|=2$
• D. no such z exists

Answer 1

The correct option is D no such z exists

Let $|z-1|=t$
Then $\frac{t+4}{t-2}<\frac{1}{2}$ ... after taking antilog $log\left(x\right)$ is decreasing in $(0,1)$.
Now
$2t+8<t-2$
$t+10<0$
$t<-10$
$|z-1|<-10$
Now $|z-1|>0$ for all z..(modulus is always greater than or equal to 0).
Hence no such z exists.