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Question

If z1, z2 and z3 are three distinct complex numbers such that 1z1z2=3z2z3=5z1z3, then the value of 1z1z2+9z2z3+25z3z1 is

A
zero
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B
1
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C
15
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D
9
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Solution

The correct option is C zero
Let 1z1z2=3z2z3=5z1z3=1k
|z1z2|=k,|z2z3|=3k,|z3z1|=5k
|z1z2|2=k2(z1z2)(¯z1¯z2)=k21z1z2=¯z1¯z2k2
|z2z3|2=9k2(z2z3)(¯z2¯z3)=9k29z2z3=¯z2¯z3k2
|z3z1|2=25k2(z3z1)(¯z3¯z1)=25k225z3z1=¯z3¯z1k2
1z1z2+9z2z3+25z3z1=¯z1¯z2k2+¯z2¯z3k2+¯z3¯z1k2=¯z1¯z2+¯z2¯z3+¯z3¯z1k2=0

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