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Question

# If z1,z2,z3 are 3 distinct complex numbers such that 3|z2âˆ’z3|=4|z3âˆ’z1|=5|z1âˆ’z2|, then the value of 9z2âˆ’z3+16z3âˆ’z1+25z1âˆ’z2 equals

A
0
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B
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C
4
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D
5
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Solution

## The correct option is A 0According tothe question:Givenrelationno:3|z2−z3|=4|z3−z1|=5|z1−z2|=k|kisconstantthen,andsquaringbothside:⇒3=k(|z2−z3|),⇒9=k2(z2−z3)2[|z|2=z¯¯¯z⇒9=k2(z2−z3)(¯¯¯¯¯z2−¯¯¯¯¯z3)−−−−−(1)⇒4=k(|z3−z1|)⇒16=k2(z3−z1)2⇒16=k2(z3−z1)(¯¯¯¯¯z3−¯¯¯¯¯z1)−−−−−(2)⇒5=k(|z1−z2|)⇒25=k2(z2−z3)2⇒25=k2(z1−z2)(¯¯¯¯¯z1−¯¯¯¯¯z2)−−−−−−(3)fromthequestion,wefindthevalue–––––––––––––––––––––––––––––––––––––––––:9z2−z3+16z3−z1+25z1−z2Nowuseingequ(1),(2)&(3)⇒k2(z2−z3)(¯¯¯¯z2−¯¯¯¯z3)(z2−z3)+k2(z3−z1)(¯¯¯¯z3−¯¯¯¯z1)(z3−z1)+k2(z1−z2)(¯¯¯¯z1−¯¯¯¯z2)(z1−z2)⇒k2(¯¯¯¯¯z2−¯¯¯¯¯z3)+k2(¯¯¯¯¯z3−¯¯¯¯¯z1)+k2(¯¯¯¯¯z1−¯¯¯¯¯z2)⇒k2¯¯¯¯¯z2−¯¯¯¯¯z3+¯¯¯¯¯z3−¯¯¯¯¯z1+¯¯¯¯¯z1−¯¯¯¯¯z2⇒k2×0∴0SothatthecorrectoptionisA.

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