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Question

If z1,z2,z3 are 3 distinct complex numbers such that 3|z2z3|=4|z3z1|=5|z1z2|, then the value of 9z2z3+16z3z1+25z1z2 equals

A
0
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B
3
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C
4
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D
5
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Solution

The correct option is A 0
According tothe question:
Givenrelationno:3|z2z3|=4|z3z1|=5|z1z2|=k|kisconstantthen,andsquaringbothside:3=k(|z2z3|),9=k2(z2z3)2[|z|2=z¯¯¯z9=k2(z2z3)(¯¯¯¯¯z2¯¯¯¯¯z3)(1)4=k(|z3z1|)16=k2(z3z1)216=k2(z3z1)(¯¯¯¯¯z3¯¯¯¯¯z1)(2)5=k(|z1z2|)25=k2(z2z3)225=k2(z1z2)(¯¯¯¯¯z1¯¯¯¯¯z2)(3)fromthequestion,wefindthevalue–––––––––––––––––––––––––––––––––––––––:9z2z3+16z3z1+25z1z2Nowuseingequ(1),(2)&(3)k2(z2z3)(¯¯¯¯z2¯¯¯¯z3)(z2z3)+k2(z3z1)(¯¯¯¯z3¯¯¯¯z1)(z3z1)+k2(z1z2)(¯¯¯¯z1¯¯¯¯z2)(z1z2)k2(¯¯¯¯¯z2¯¯¯¯¯z3)+k2(¯¯¯¯¯z3¯¯¯¯¯z1)+k2(¯¯¯¯¯z1¯¯¯¯¯z2)k2¯¯¯¯¯z2¯¯¯¯¯z3+¯¯¯¯¯z3¯¯¯¯¯z1+¯¯¯¯¯z1¯¯¯¯¯z2k2×00SothatthecorrectoptionisA.

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