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Question Number 21219 by Tinkutara last updated on 16/Sep/17

$$\left(\mathrm{A}\right)\mathrm{If}\mid w\mid =2,\mathrm{then}\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{points}$$$$z=w-\frac{1}{w}\mathrm{is}\mathrm{contained}\mathrm{in}\mathrm{or}\mathrm{equal}\mathrm{to}$$$$\left(\mathrm{B}\right)\mathrm{If}\mid w\mid =1,\mathrm{then}\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{points}$$$$z=w+\frac{1}{w}\mathrm{is}\mathrm{contained}\mathrm{in}\mathrm{or}\mathrm{equal}\mathrm{to}$$$$\mathrm{Options}\mathrm{for}\mathrm{both}\mathrm{A}\mathrm{and}\mathrm{B}:$$$$\left(\mathrm{p}\right)\mathrm{An}\mathrm{ellipse}\mathrm{with}\mathrm{eccentricity}\frac{4}{5}$$$$\left(\mathrm{q}\right)\mathrm{The}\mathrm{set}\mathrm{of}\mathrm{points}z\mathrm{satisfying}\mathrm{Im}z$$$$=0$$$$\left(\mathrm{r}\right)\mathrm{The}\mathrm{set}\mathrm{of}\mathrm{points}z\mathrm{satisfying}\mid \mathrm{Im}z\mid$$$$⩽1$$$$\left(\mathrm{s}\right)\mathrm{The}\mathrm{set}\mathrm{of}\mathrm{points}z\mathrm{satisfying}\mid \mathrm{Re}z\mid$$$$⩽2$$$$\left(\mathrm{t}\right)\mathrm{The}\mathrm{set}\mathrm{of}\mathrm{points}z\mathrm{satisfying}\mid z\mid ⩽3$$

Answered by Tinkutara last updated on 17/Sep/17

$$\left(\mathbf{A}\right)Letw=2(\cos \theta +i\sin \theta )$$$$Then\frac{1}{w}=\frac{1}{2}(\cos \theta -i\sin \theta )$$$$w-\frac{1}{w}=\frac{3}{2}\cos \theta +\frac{5}{2}i\sin \theta =x+iy(=z)$$$$\Rightarrow \cos \theta =\frac{2x}{3};\sin \theta =\frac{2y}{5}$$$$\frac{4x^2}{9}+\frac{4y^2}{25}=1\Rightarrow Ellipsewithe=\frac{4}{5}$$$$\left(\mathbf{B}\right)w=\cos \theta +i\sin \theta ;\frac{1}{w}=\cos \theta -i\sin \theta$$$$z=2\cos \theta$$$$\therefore \mid z\mid ⩽2andIm\left(z\right)=0.$$

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