If `A (z_1), B (z_2) and C (z_3)` are three points in the argand plane where `|z_1 +z_2|=||z_1-z_2| and |(1-i)z_1+iz_3|=|z_1|+|z_3|-z_1|`, where `i = sqrt-1` then

Options

A. A,B and C lie on a circle with center `(z_(2)+z_(3))/(2)`
B. A,B and C are collinear points.
C. A,B,C from an equilateral triangle.
D. A,B,C form an obtuse angle triangle.

Answer

A

Explanation

We have,

`|z_(1)+z_(2)|=||z_(1)|-|z_(2)|| ltimplies "Arg" (z_(1)/z_(2))=+-pi`
and `|(1-i)z_(1)+iz_(3)|=|z_(1)|+|z_(3)-z_(1)|`
`ltimplies |z_(1)+i(z_(3)-z_(1))|=|z_(1)|+|z_(3)-z_(1)|`
`rArr "arg"(z_(1)/(z_(3)-z_(1))=pi/2 [therefore |z_(1)+iz_(2)|=|z_(1)|+|z_(2)| ltimplies "arg"(z_(1)/z_(2))=pi/2]`
Thus, A lies on the circle with BC as diameter i.e., `(z_(2)+z_(3))/(2)` as center. Hence, option (a) is correct.