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Question

Let z1,z2,z3 be three complex numbers and a,b,c be real numbers not all zero, such that a+b+c=0 and az1+bz2+cz3=0, then

A
z1,z2,z3 are vertices of a triangle.
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B
z1,z2,z3 lies on circumference of a circle
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C
z1,z2,z3 are collinear.
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D
None of these.
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Solution

The correct option is C z1,z2,z3 are collinear.
Given,
a+b+c=0....(i)
and
az1+bz2+cz3=0....(ii)
Since a,b,c are not all zero, from (i) and (ii), we have
az1+bz2(a+b)z3=0
az1+bz2=(a+b)z3
z3=az1+bz2a+b....(iii)
From (iii), it follows that z3 divides the line segment joining z1 and z2 internally in the ratio b:a. If a and b are of opposite sign, then division is external in the ratio |b|:|a|.

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