Question

The locus of the complex number z is an argand plane satisfying the inequality $lo{g}_{\frac{1}{2}}\left(\frac{|z-1|+4}{3|z-1|-2}\right)>1(where|z-1|\ne \frac{2}{3})$

• A. a circle
• B. interior of a circle
• C. exterior of a circle
• D. None of these

Answer 1

The correct option is C exterior of a circle

We have , $lo{g}_{\frac{1}{2}}\left(\frac{|z-1|+4}{3|z-1|-2}\right)>lo{g}_{\frac{1}{2}}\left(\frac{1}{2}\right)lo{g}_{\frac{1}{2}}\left(\frac{|z-1|+4}{3|z-1|-2}\right)>lo{g}_{\frac{1}{2}}\left(\frac{1}{2}\right)\Rightarrow \frac{|z-1|+4}{3|z-1|-2}<\frac{1}{2}<1[\because lo{g}_{a}xisadecreasingfunctionifa<1]\Rightarrow |z-1|+4<3|z-1|-2\Rightarrow 2|z-1|>6\Rightarrow |z-1|>3$