Question

Let $C_1$ and $C_2$ be concentric circles of radius $1$ and $\frac{8}{3}$ respectively, having centre at $(3,0)$ on the Argand plane. If the complex number $z$ satisfies the inequality ${\log }_{\frac{1}{3}}\left(\frac{|z-3{|}^2+2}{11|z-3|-2}\right)>1$, then

• A. $z$ lies outside $C_1$ but inside $C_2$
• B. $z$ lies inside both $C_1$ and $C_2$
• C. $z$ lies outside both $C_1$ and $C_2$
• D. none of these

Answer 1

The correct option is A $z$ lies outside $C_1$ but inside $C_2$

${\log }_{\frac{1}{3}}\left(\frac{|z-3{|}^2+2}{11|z-3|-2}\right)>1$.
$\Rightarrow \frac{|z-3{|}^2+2}{11|z-3|-2}<\frac{1}{3}$
$\Rightarrow (3t-8)(t-1)<0$
(where $|z-3|=t$)
$\Rightarrow 1<|z-3|<\frac{8}{3}$
Hence, $z$ lies outside $C_1$ but inside $C_2$