Let `z_(1)` and `z_(2)` be two distinct complex numbers and `z=(1-t)z_(1)+tz_(2)`, for some real number t with `0 lt t lt 1` and `i=sqrt(-1)`. If arg(w) denotes the principal argument of a non-zero compolex number w, then

Options

A. `|z-z_(2)|+|z-z_(2)|=|z_(1)-z_(2)|`
B. `"arg"(z-z_(1))="arg"(z-z_(2))`
C. `|{:(z-z_(1),barz-barz_(1)),(z_(2)-z_(1),barz_(2)-barz_(1)):}|=0`
D. `"arg"(z-z_(1))="arg"(z_(2)-z_(1))`

Answer

B

Explanation

Let `z_(1)` and `z_(2)` be represented by points A and B respectively in the Argand plane.

We have,
`z=((1-t)z_(1)+tz_(2))/((1-t)+t)`
`rArr` z represents a point P dividing AB in the ratio `t:(1-t)`
Clearly, AP+PB=AB
`rArr |z-z_(1)|+|z-z_(2)|=|z_(1)-z_(2)|`
So, option (a) is true.
Since, A,P,B are collinear. Therefore,
`"arg"(barAP)="arg"(barAB) rArr "arg"(z-z_(1))="arg"(z_(2)-z_(1))`
So, option (d) is correct.
We have,
`z=(1-t)z_(1)+tz_(2)`
`rArr t=(z-z_(1))/(z_(2)-z_(1))` is purely real
`rArr |{:(z-z_(1),barz-barz_(1)),(z_(2)-z_(1),barz_(2),barz_(1)):}|=0`
Hence, option (c) is correct.
We have,
`"arg"(z-z_(1))="arg"(barAP)` and `"arg"(z-z-(2))="arg"(barBP)`
Clearly, `"arg"(barAP)-"arg"(z-z_(2)) ne 0`
So, option (b) is not correct.