1
Question
If |z1|=1,|z2|=2,|z3|=3, and |9z1z2+4z1z3+z2z3|=12, Then find the value of |z1+z2+z3|.
If |z1|=1,|z2|=2,|z3|=3, and |9z1z2+4z1z3+z2z3|=12, Then find the value of |z1+z2+z3|.
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Solution
The correct option is A 2,
|z1|=1⟹z1¯¯¯¯¯z1=1,|z2|=2⟹z2¯¯¯¯¯z2=4,|z3|=3⟹z3¯¯¯¯¯z3=9
Also,
|9z1z2+4z1z3+z2z3|=12
or ∣∣z1z2z3¯¯¯z3+z1z2z3¯¯¯z2+z1¯¯¯z1z2z3∣∣=12
or |z1z2z3|∣∣¯¯¯z1+¯¯¯z2+¯¯¯z3∣∣=12
or |z1||z2||z3|∣∣¯¯¯z1+¯¯¯z2+¯¯¯z3∣∣=12
or 6∣∣¯¯¯z1+¯¯¯z2+¯¯¯z3∣∣=12
or ∣∣¯¯¯z1+¯¯¯z2+¯¯¯z3∣∣=2
or |z1+z2+z3|=2
Ans: A
|z1|=1⟹z1¯¯¯¯¯z1=1,|z2|=2⟹z2¯¯¯¯¯z2=4,|z3|=3⟹z3¯¯¯¯¯z3=9
Also,
|9z1z2+4z1z3+z2z3|=12
or ∣∣z1z2z3¯¯¯z3+z1z2z3¯¯¯z2+z1¯¯¯z1z2z3∣∣=12
or |z1z2z3|∣∣¯¯¯z1+¯¯¯z2+¯¯¯z3∣∣=12
or |z1||z2||z3|∣∣¯¯¯z1+¯¯¯z2+¯¯¯z3∣∣=12
or 6∣∣¯¯¯z1+¯¯¯z2+¯¯¯z3∣∣=12
or ∣∣¯¯¯z1+¯¯¯z2+¯¯¯z3∣∣=2
or |z1+z2+z3|=2
Ans: A
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