The-area-of-the-region-in-the-complex-plane-satisfying-the-inequality-log-cos-pi-6-z-2-5-4-z-2-4-lt-2-is-

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

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

Answered by MJS_new last updated on 22/Apr/21

$$\mid{z}−\mathrm{2}\mid={x}\geqslant\mathrm{0} \\ $$$$\frac{\mathrm{ln}\:\frac{{x}+\mathrm{5}}{\mathrm{4}\left({x}−\mathrm{1}\right)}}{\mathrm{ln}\:\mathrm{cos}\:\frac{\pi}{\mathrm{6}}}<\mathrm{2}\:\Leftrightarrow\:\mathrm{ln}\:\frac{{x}+\mathrm{5}}{{x}−\mathrm{1}}\:>\mathrm{ln}\:\mathrm{3} \\ $$$$\frac{{x}+\mathrm{5}}{{x}−\mathrm{1}}>\mathrm{3}\:\Rightarrow\:{x}<\mathrm{4} \\ $$$$\mathrm{0}\leqslant{x}<\mathrm{4} \\ $$$$\mathrm{0}\leqslant\mid{z}−\mathrm{2}\mid<\mathrm{4} \\ $$$$\mathrm{this}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{center}\:\begin{pmatrix}{\mathrm{2}}\\{\mathrm{0}}\end{pmatrix}\:\mathrm{and}\:\mathrm{radius}\:\mathrm{4} \\ $$$$\mathrm{but}\:\mathrm{without}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{line} \\ $$

Commented by EnterUsername last updated on 22/Apr/21

$${Thank}\:{you}\:{Sir} \\ $$

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