1
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Question

# If z1, z2 and z3 are three distinct complex numbers such that 1âˆ£z1âˆ’z2âˆ£=3âˆ£z2âˆ’z3âˆ£=5âˆ£z1âˆ’z3âˆ£, then the value of 1z1âˆ’z2+9z2âˆ’z3+25z3âˆ’z1 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 zeroLet 1∣z1−z2∣=3∣z2−z3∣=5∣z1−z3∣=1k|z1−z2|=k,|z2−z3|=3k,|z3−z1|=5k⟹|z1−z2|2=k2⟹(z1−z2)(¯z1−¯z2)=k2⟹1z1−z2=¯z1−¯z2k2⟹|z2−z3|2=9k2⟹(z2−z3)(¯z2−¯z3)=9k2⟹9z2−z3=¯z2−¯z3k2⟹|z3−z1|2=25k2⟹(z3−z1)(¯z3−¯z1)=25k2⟹25z3−z1=¯z3−¯z1k21z1−z2+9z2−z3+25z3−z1=¯z1−¯z2k2+¯z2−¯z3k2+¯z3−¯z1k2=¯z1−¯z2+¯z2−¯z3+¯z3−¯z1k2=0

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