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

Let A, B, C be three sets of complex numbers as defined below A = {z : Im z ≥ 1} B = {z : ∣z − 2 − i∣ = 3} C = {z : Re((1 − i)z) = (√2)}. Let z be any point in A ∩ B ∩ C and let w be any point satisfying ∣w − 2 − i∣ < 3. Then, ∣z∣ − ∣w∣ + 3 lies between (1) −6 and 3 (2) −3 and 6 (3) −6 and 6 (4) −3 and 9

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

Answered by Tinkutara last updated on 15/Sep/17

Given that ∣w−2−i∣<3 ⇒Distance between w and 2+i i.e. S is smaller than 3. ⇒w is a point lying inside the circle with centre S and radius 3. ⇒Distance between z (i.e. the point P) and w should be smaller than 6 (the diameter of the circle) i.e. ∣z−w∣<6 But we know that ∣∣z∣−∣w∣∣<∣z−w∣ ⇒∣∣z∣−∣w∣∣<6⇒−6<∣z∣−∣w∣<6 −3<∣z∣−∣w∣+3<9

$${Given}\:{that}\:\mid{w}−\mathrm{2}−{i}\mid<\mathrm{3} \\ $$$$\Rightarrow{Distance}\:{between}\:{w}\:{and}\:\mathrm{2}+{i}\:{i}.{e}.\:{S} \\ $$$${is}\:{smaller}\:{than}\:\mathrm{3}. \\ $$$$\Rightarrow{w}\:{is}\:{a}\:{point}\:{lying}\:{inside}\:{the}\:{circle} \\ $$$${with}\:{centre}\:{S}\:{and}\:{radius}\:\mathrm{3}. \\ $$$$\Rightarrow{Distance}\:{between}\:{z}\:\left({i}.{e}.\:{the}\:{point}\:{P}\right) \\ $$$${and}\:{w}\:{should}\:{be}\:{smaller}\:{than}\:\mathrm{6}\:\left({the}\right. \\ $$$$\left.{diameter}\:{of}\:{the}\:{circle}\right) \\ $$$${i}.{e}.\:\mid{z}−{w}\mid<\mathrm{6} \\ $$$${But}\:{we}\:{know}\:{that}\:\mid\mid{z}\mid−\mid{w}\mid\mid<\mid{z}−{w}\mid \\ $$$$\Rightarrow\mid\mid{z}\mid−\mid{w}\mid\mid<\mathrm{6}\Rightarrow−\mathrm{6}<\mid{z}\mid−\mid{w}\mid<\mathrm{6} \\ $$$$−\mathrm{3}<\mid{z}\mid−\mid{w}\mid+\mathrm{3}<\mathrm{9} \\ $$

Commented by Tinkutara last updated on 15/Sep/17

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