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 . Then, | z + 1 - i | 2 + | z - 5 - i | 2 lies between

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. $Let z be any point in A \cap B \cap C .$ $Then,$ $| z + 1 - i |^{2}$ $+$ $| z - 5 - i |^{2}$ $lies between$ [2008]

Smart Achievers · Accepted Answer

(3)

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

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

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

$\Rightarrow$ 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:

$Since, z is a point of A \cap B \cap C \Rightarrow z represents the point P$

$∴$ $| z + 1 - i |^{2} + | z - 5 - i |^{2}$

$\Rightarrow$ $| z - (- 1 + i) |^{2} + | z - (5 + i) |^{2}$

$\Rightarrow P Q^{2} + P R^{2} = Q R^{2} = 6^{2} = 36, which lies between 35 and 39$

$∴$ $(3) is correct option.$

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