Question

The locus of the complex number $z$ in an argand plane satisfying the inequality ${\log }_{1/2}\left(\frac{|z-1|+4}{3|z-1|-2}\right)>1($ where $|z-1|\ne \frac{2}{3})$ is

• A. A circle
• B. Interior of a circle
• C. Exterior of a circle
• D. None of these

Answer 1

Answer: (C)

Exterior of a circle

We have, ${\log }_{1/2}\left(\frac{|z-1|+4}{3|z-1|-2}\right)>1={\log }_{1/2}\left(\frac{1}{2}\right)$

$\Rightarrow \frac{|z-1|+4}{3|z-1|-2}<\frac{1}{2}<1$

$(\because {\log }_{a}x$ is a decreasing function if $a<1)$

$\Rightarrow |z-1|+4<3|z-1|-2\Rightarrow 2|z-1|>6\Rightarrow |z-1|>3$

which is exterior of a circle