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Question Number 21108 by Tinkutara last updated on 13/Sep/17

$$\mathrm{Let}A,B,C\mathrm{be}\mathrm{three}\mathrm{sets}\mathrm{of}\mathrm{complex}$$$$\mathrm{numbers}\mathrm{as}\mathrm{defined}\mathrm{below}$$$$A=\{z:\mathrm{Im}z⩾1\}$$$$B=\{z:\mid z-2-i\mid =3\}$$$$C=\{z:\mathrm{Re}\left((1-i)z\right)=\sqrt{2}\}.$$$$\mathrm{Let}z\mathrm{be}\mathrm{any}\mathrm{point}\mathrm{in}A\cap B\cap C\mathrm{and}\mathrm{let}$$$$w\mathrm{be}\mathrm{any}\mathrm{point}\mathrm{satisfying}\mid w-2-i\mid <$$$$3.\mathrm{Then},\mid z\mid -\mid w\mid +3\mathrm{lies}\mathrm{between}$$$$\left(1\right)-6\mathrm{and}3$$$$\left(2\right)-3\mathrm{and}6$$$$\left(3\right)-6\mathrm{and}6$$$$\left(4\right)-3\mathrm{and}9$$

Answered by Tinkutara last updated on 15/Sep/17

$$Giventhat\mid w-2-i\mid <3$$$$\Rightarrow Distancebetweenwand2+ii.e.S$$$$issmallerthan3.$$$$\Rightarrow wisapointlyinginsidethecircle$$$$withcentreSandradius3.$$$$\Rightarrow Distancebetweenz(i.e.thepointP)$$$$andwshouldbesmallerthan6(the$$$$diameterofthecircle)$$$$i.e.\mid z-w\mid <6$$$$Butweknowthat\mid \mid z\mid -\mid w\mid \mid <\mid z-w\mid$$$$\Rightarrow \mid \mid z\mid -\mid w\mid \mid <6\Rightarrow -6<\mid z\mid -\mid w\mid <6$$$$-3<\mid z\mid -\mid w\mid +3<9$$

Commented by Tinkutara last updated on 15/Sep/17

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