Question

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

• A. $z$ lies outside $C_1$ but inside $C_2$
• B. $z$ lies inside of 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 $C_2$

${\log }_{1/3}\left(\frac{{|z-3|}^2+2}{11|z-3|-2}\right)=>\left(\frac{{|z-3|}^2+2}{11|z-3|-2}\right)<\frac{1}{3}$
Let $|z-3|=k$
Thus, $\frac{k^2+2}{11k-2}<\frac{1}{3}=>3k^2-11k+8<0=>(3k-8)(k-1)<0=>1<k<\frac{8}{3}$
Thus, $1<|z-3|<\frac{8}{3}$
Hence, (a) is correct.