Question

If $\log _{1 / 2}\left(\frac{|z-1|+4}{3|z-1|-2}\right)>1$, then the locus of $z$ is

• A. Exterior to circle with center $1+i 0$ and radius 10
• B. Interior to circle with center $1+i 0$ and radius 10
• C. Circle with center $1+i 0$ and radius 10
• D. Circle with center $2+i 0$ and radius 10

Answer 1

Answer: (A)

$\log _{1 / 2}\left(\frac{|z-1|+4}{3|z-1|-2}\right)>1 $
$0<\frac{|z-1|+4}{3|z-1|-2}<\frac{1}{2} |z-1|=t $
$0<\frac{t+4}{3 t-2}<\frac{1}{2} $
$0<\frac{t+4}{3 t-2}<\frac{1}{2} \Rightarrow t>10$
So true