Q. Let z be any point A B C and let w be any point satisfying | w - 2 - i | 3 . Then, | z | - | w | + 3 lies between [2008]

Question

$Let A, B, C be three sets of complex numbers as defined below$

$A = {z : Im (z) \geq 1}$

$B = {z : | z - 2 - i | = 3}$

$C = {z : Re ((1 - i) z) = \sqrt{2}}$

Q. $Let z be any point A \cap B \cap C and let w be any point satisfying$ $| w - 2 - i |$ $< 3 .$ $Then, | z | - | w | + 3 lies between$ [2008]

Smart Achievers · Accepted Answer

(4)

$Given: A = {z : Im (z) \geq 1} = {(x, y) : y \geq 1}$

$Clearly A is the set of all points lying on or above the line y = 1 in Cartesian plane.$

$B = {z : | z - 2 - i | = 3} = {(x, y) : {(x - 2)}^{2} + {(y - 1)}^{2} = 9}$

$\Rightarrow$ B is the set of all points lying on the boundary of the circle with centre (2, 1) and radius 3.

$C = {z : Re [(1 - i) z] = \sqrt{2}} = {(x, y) : x + y = \sqrt{2}}$

$\Rightarrow$ C is the set of all points lying on the straight line represented by $x + y = \sqrt{2} .$

Graphically, the three sets are represented as shown below:

$Given$ $| w - 2 - i |$ $< 3$

$\Rightarrow Distance between w and (2 + i), i.e. S is smaller than 3 .$

$\Rightarrow w is a point lying inside the circle with centre S and radius 3 .$

$\Rightarrow Distance between z (i.e. the point P) and w should be smaller than 6 (the diameter of the circle)$

$i.e. | z - w | < 6$

$But we know that$ $| | z | - | w | |$ $\leq | z - w |$

$\Rightarrow$ $| | z | - | w | |$ $< 6 \Rightarrow - 6 < | z | - | w | < 6$

$\Rightarrow - 3 < | z | - | w | + 3 < 9$

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