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Question

# Let z1 and z2 be two complex numbers represented by points on the circle |z1|=1 and |z2|=2 respectively. Let P be the point whose coordinates are (1,1). Then

A
the maximum value of |2z1+z2| is 4.
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B
the minimum value of |z1z2| is 1.
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C
z2+1z13
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D
the minimum distance of P from z1 is 22 unit.
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Solution

## The correct option is C ∣∣∣z2+1z1∣∣∣≤3|2z1+z2|≤2|z1|+|z2|=2×1+2=4 ∴ Maximum value of |2z1+z2| is 4. |z1−z2| is least when 0,z1,z2 are collinear. |z1−z2|≥∣∣|z1|−|z2|∣∣=|1−2|=1 Then min|z1−z2|=1 Again, ∣∣∣z2+1z1∣∣∣≤|z2|+∣∣∣1z1∣∣∣=|z2|+1|z1|=2+11=3⇒∣∣∣z2+1z1∣∣∣≤3 P lies outside |z1|=1 Hence, required minimum distance =OP−r, where O≡(0,0) =√2−1

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