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Question

# Assertion :Let z1,z2,z3 be three complex numbers such that |3z1+1|=|3z2+1|=|3z3+1| and 1+z1+z2+z3=0, then z1,z2,z3 will represent vertices of an equilateral triangle on the complex plane. Reason: z1,z2,z3 represent vertices of an equilateral triangle if z21+z22+z23=z1z2+z2z3+z3z1.

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Assertion is incorrect but Reason is correct
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Solution

## The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for AssertionAssertion: |3z1+1|=|3z2+1|=|3z3+1|Hence, ∣∣∣z1−(−13)∣∣∣ =∣∣∣z2−(−13)∣∣∣=∣∣∣z3−(−13)∣∣∣Hence, −13 is the circumcentre of the triangle formed by z1,z2 and z3. 1+z1+z2+z3=0. ⇒z1+z2+z33=−13Hence, −13 is also the centroid of the triangle formed by z1,z2 and z3.As the centroid and the circumcentre are the same, the triangle is a equilateral triangle. Hence, the Assertion is true.Reason: z21+z22+z23=z1z2+z2z3+z3z1.or z21+z22+z23−z1z2−z2z3−z3z1=0 ....(1)Let A(z1),B(z2),C(z3) represents the vertices of △ABCSince, △ABC is equilateral, the vector BC can be obtained by rotating AB anti-clockwise through 1200 (z3z2)=(z2z1)ei2/3(z3z2)=(z2z1)ω⇒z1ωz2ωz2+z3=0Multiplying by ω2 z1z2z2ω2+z3ω2=0z1(1+ω2)z2+ω2z3=0⇒z1=−ωz2−ω2z3 ....(2)Now ,consider LHS=z21+z22+z23−z1z2−z2z3−z3z1Put the value of z1 from eqn (2),=(ωz2+ω2z3)2+z22+z23+(ωz2+ω2z3)z2−z2z3−z3(ωz2+ω2z3)=z22(1+ω+ω2)+z22(1+ω+ω2)+z2z3z22(1+ω+ω2)=0+0+0=0Hence, reason is true.However, the reason does not explain the assertion.

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