Plot the region represented by `pi/3lt=a r g((z+1)/(z-1))lt=(2pi)/3` in the Argand plane.

Find the center of the are represented by a r g[(z-3i)//(z-2i+4)]=pi//4 .

if arg(z+a)=pi/6 and arg(z-a)=(2pi)/3 then

If the vertices of an equilateral triangle be represented by the complex numbers z_(1),z_(2),z_(3) on the Argand diagram, then prove that, z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1).

Let a , b , c be distinct complex numbers with |a|=|b|=|c|=1 and z_(1) , z_(2) be the roots of the equation az^(2)+bz+c=0 with |z_(1)|=1 . Let P and Q represent the complex numbers z_(1) and z_(2) in the Argand plane with /_POQ=theta , o^(@) lt 180^(@) (where O being the origin).Then

The area bounded by the curves arg z = pi/3 and arg z = 2 pi /3 and arg(z-2-2isqrt3) = pi in the argand plane is (in sq. units)

Prove that |z-z_1|^2+|z-z_2|^2=k will represent a real circle with center ((z_1+z_2)/2) on the Argand plane if 2kgeq|z_1-z_2|^2

Let z_(1) be a fixed point on the circle of radius 1 centered at the origin in the Argand plane and z_(1)ne+-1 . Consider an equilateral traingle inscribed in the circle with z_(1),z_(2),z_(3) as the vertices taken in the counter clockwise direction .Then z_(1)z_(2)z_(3) is equal to -

Let z_1 be a fixed point on the circle of radius 1 centered at the origin in the Argand plane and z_1 ne+-1 consider an equilateral triangle inscribed in the circle with z_1,z_2,z_3 as the vertices taken in the counter clockwise directtion then z_1z_2z_3 is equal to

The points representing the complex number z for which arg ((z-2)/(z+2))=(pi)/(3) lie on -

The points representing the complex number z for which arg((z-2)/(z+2))=pi/3 lie on