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  4. If log 1/2(|z|2+2|z|+4/2|z|2+1)<0, then the region traced by z is

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Q. If $ {{\log }_{1/2}}\frac{|z{{|}^{2}}+2|z|+4}{2|z{{|}^{2}}+1}<0, $ then the region traced by z is

JamiaJamia 2013

Solution:

$ \because $ $ {{\log }_{1/2}}\frac{|z{{|}^{2}}+2|z|+4}{2|z{{|}^{2}}+1}<0={{\log }_{1/2}}1 $ $ \Rightarrow $ $ \frac{|z{{|}^{2}}+2|z|+4}{2|z{{|}^{2}}+1}<1 $ $ (\because base<1) $ $ \Rightarrow $ $ |z{{|}^{2}}-2|z|-3<0 $ $ (|z|+1)(|z|-3)<0 $ $ \therefore $ $ |z|<3 $