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## Meinme Group Title The locus of Z satisfying the inequality given below is one year ago one year ago

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1. Meinme

$\log_{1/3} \left| z+1 \right|>\log_{1/3}\left| z-1 \right|$

2. Meinme

options are a) R(z) < 0 b) R(z) > 0 c) I(z) < 0 d) None of these Please provide step wise procedure with answer

3. vf321

$\log_{1/3}|z+1|> \log_{1/3}|z-1|$Since $$e^x$$ is monotonic increasing over the real domain $$(\frac{1}{3})^x$$ is monotonic decreasing over all reals. Thus, $(1/3)^{\log_{1/3}|z+1|} < (1/3)^{\log_{1/3}|z-1|}$$|z+1| < |z -1|$For a complex number $$z = a + b i$$, the only way the magnitude can be decreased by increasing the real component is if $$a$$ is negative, because then the $$(a-1)^2$$ term in the magnitude $$\sqrt{(a-1)^2+b^2}$$ increases, as opposed to if it was $$a+1$$. Thus, the answer is (a)