If complex number z satisfies the inequality |z−1−2i|≤1, then which of the following option(s) is (are) CORRECT ?
A
maximum value of |z| is √5+1
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B
minimum value of |z| is √5
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C
maximum value of arg(z) is π2
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D
minimum value of arg(z) is tan−1(34)
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Solution
The correct options are A maximum value of |z| is √5+1 C maximum value of arg(z) is π2 D minimum value of arg(z) is tan−1(34) |z−1−2i|≤1 represents the points inside and on the circle with centre C(1,2) and radius r=1
∴|z|min=OC−r=√12+22−1=√5−1 and |z|max=OC+r=√12+22+1=√5+1
Clearly, arg(z) is maximum for M(z). max{arg(z)}=π2 arg(z) is minimum for N(z). min{arg(z)}=θ Now, tanθ=tan(π2−2α) =cot2α =1tan2α =1−tan2α2tanα =1−(12)22×12=34 ∴θ=tan−134