Question

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

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

Answer 1

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

${\log }_{1/3}\left(\frac{|z-3{|}^2+2}{11|z-3|-2}\right)>1$
Let $|z-3|=t\Rightarrow t>0$
for function to be defined
$\frac{t^2+2}{11t-2}>0$
$\Rightarrow t>\frac{2}{11}\cdots \left(1\right)$

${\log }_{1/3}\left(\frac{|z-3{|}^2+2}{11|z-3|-2}\right)>1$
$\Rightarrow \frac{t^2+2}{11t-2}<\frac{1}{3}$
$\Rightarrow \frac{3t^2-11t+8}{3(11t-2)}<0$
$\Rightarrow \frac{(3t-8)(t-1)}{11t-2}<0$
$\Rightarrow t\in (1,\frac{8}{3})\cdots \left(2\right)\left[\text{using}\right(1\left)\right]$
Hence
$1<t<\frac{8}{3}$
$\Rightarrow 1<|z-3|<\frac{8}{3}$
$\therefore z$ lies outside $C_1$ but inside $C_2$