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Question Number 139057 by EnterUsername last updated on 21/Apr/21

$$\mathrm{The}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{region}\mathrm{in}\mathrm{the}\mathrm{complex}\mathrm{plane}\mathrm{satisfying}$$ $$\mathrm{the}\mathrm{inequality}{\log }_{\cos \left(\frac{\pi }{6}\right)}\left[\frac{\mid \mathrm{z}-2\mid +5}{4\mid \mathrm{z}-2\mid -4}\right]<2\mathrm{is}?$$

Answered by MJS_new last updated on 22/Apr/21

$$\mid z-2\mid =x⩾0$$ $$\frac{\ln \frac{x+5}{4(x-1)}}{\ln \cos \frac{\pi }{6}}<2\iff \ln \frac{x+5}{x-1}>\ln 3$$ $$\frac{x+5}{x-1}>3\Rightarrow x<4$$ $$0⩽x<4$$ $$0⩽\mid z-2\mid <4$$ $$\mathrm{this}\mathrm{is}\mathrm{a}\mathrm{circle}\mathrm{with}\mathrm{center}\left(\begin{array}{c}2\\ 0\end{array}\right)\mathrm{and}\mathrm{radius}4$$ $$\mathrm{but}\mathrm{without}\mathrm{the}\mathrm{circle}\mathrm{line}$$

Commented byEnterUsername last updated on 22/Apr/21

$$ThankyouSir$$

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