Question

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

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

Answer 1

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

Let $|z-3|=t$
Hence the above in-equation reduces to
$\frac{t^2+2}{11t-2}<\frac{1}{3}$
$3t^2+6<11t-2$
$3t^2-11t+8<0$
$(t-1)(3t-8)<0$
Hence, $1<t<\frac{8}{3}$
$1<|z-3|<\frac{8}{3}$

Hence, $z$ lies between two concentric circles.