Question

Let $C_1$ and $C_2$ are concentric circles of radius $1$ and $8/3$ respectively, having centre at $(3,0)$ on the Argand plane. If the complex $z$ 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 of both $C_1$ and $C_2$
• C. $z$ lies outside of both $C_1$ and $C_2$
• D. none of these

Answer 1

Answer: (A)

$z$ lies outside $C_1$ but inside $C_2$

Solution:

We can consider $p=|z-3|$.

So we can apply P in log values.

$\log \frac{1}{3}\frac{p^2+2}{11p-2}>1$

$\frac{p^2+2}{11p-2}>0$ here

numerator is always positive, So $11p-2>0$

then $p>\frac{2}{11}$

$\frac{p^2+2}{11p-2}<\frac{1}{3}$

$\frac{p^2+2}{11p-2}-\frac{1}{3}$

$\frac{{3p}^2+6-11p+2}{3(11p-2)}<0$

$\frac{{3p}^2-11p+8}{3(11p-2)}<0$

The denominator is positive for the previous condition, then the numerator must be negative

$1<|z-3|<\frac{8}{3}$

Answer: z is outside $C_1$ and inside $C_2$