IIT-JEE 2008 · Paper 1 · Mathematics Q22

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MathematicsSingle correct+4 / -1Question 22 of 69

Passage

Paragraph for Question Nos. 21 to 23

Let $A$ , $B$ , $C$ be three sets of complex numbers as defined below
$A=\{z:\operatorname{Im}z\ge1\},\qquad B=\{z:|z-2-i|=3\},\qquad C=\{z:\operatorname{Re}((1-i)z)=\sqrt2\}.$

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

25 and 29

30 and 34

35 and 39

40 and 44

Answer and worked solution

Official answer: C

Diameter. The points $-1+i$ and $5+i$ , i.e. $(-1,1)$ and $(5,1)$ , are the ends of the horizontal diameter of circle B.

Right angle. Any $z$ on B (in particular the point of $A\cap B\cap C$ ) sees that diameter at a right angle, so
$|z+1-i|^2+|z-5-i|^2=(\text{diameter})^2=36,$
which lies between 35 and 39.

(C)

Topics:Geometry in the complex plane

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IIT-JEE 2008 · Paper 1, Mathematics question 22