CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
0
You visited us 0 times! Enjoying our articles? Unlock Full Access!
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
No worries! We‘ve got your back. Try BYJU‘S free classes today!
B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
C
Assertion is correct but Reason is incorrect
No worries! We‘ve got your back. Try BYJU‘S free classes today!
D
Assertion is incorrect but Reason is correct
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
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+z23z1z2z2z3z3z1=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=ωz2ω2z3 ....(2)
Now ,consider LHS=z21+z22+z23z1z2z2z3z3z1
Put the value of z1 from eqn (2),
=(ωz2+ω2z3)2+z22+z23+(ωz2+ω2z3)z2z2z3z3(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.

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Introduction
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon