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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.
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.
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Solution
The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
Assertion:
|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=−13
Hence, −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 △ABC
Since, △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=0
Multiplying by ω2
z1z2z2ω2+z3ω2=0
z1(1+ω2)z2+ω2z3=0
⇒z1=−ωz2−ω2z3 ....(2)
Now ,consider LHS=z21+z22+z23−z1z2−z2z3−z3z1
Put 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=0
Hence, reason is true.
However, the reason does not explain the assertion.
Assertion:
|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=−13
Hence, −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 △ABC
Since, △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=0
Multiplying by ω2
z1z2z2ω2+z3ω2=0
z1(1+ω2)z2+ω2z3=0
⇒z1=−ωz2−ω2z3 ....(2)
Now ,consider LHS=z21+z22+z23−z1z2−z2z3−z3z1
Put 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=0
Hence, reason is true.
However, the reason does not explain the assertion.
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