Question

If three complex numbers are in A.P., then they lie on [IIT \[1985\]; DCE \[2001\]; Pb. CET \[2003\]]
 A circle in the complex plane
A straight line in the complex plane
A parabola in the complex plane
None of these

Answer 1

Hint: in this question we have to find locus of three complex number which are in AP. Simply use property of AP which state twice the middle term of AP is sum of first term and third term.

Formula Used: If a,b and c are in AP then
\[2b = a + c\]
Where
a,b and c are any number or variable

Complete step by step solution: Given: Three complex numbers are in AP
Let three complex numbers be \[{z_1},{z_2}\]and\[{z_3}\]
Now we know that
If a,b and c are in AP then
\[2b = a + c\]
Where
a,b and c are any number or variable
It is given that three complex numbers are in AP
\[2{z_2} = {z_1} + {z_3}\]
By using above equation we can say that \[{z_2}\] is mid point line having \[{z_1}\] and \[{z_3}\]as its end point
So, these three complex number lies in a straight line.

Option ‘B’ is correct

Note: We must remember that twice the middle term of AP is sum of first term and third term and midpoint of any line divide it in two equal parts.
Don’t try to put complex number z in the form of real and imaginary form because it makes solution complicated and sometime you will not get correct answer in these types of questions.