If `z=z_0+A( bar z -( bar z _0)), w h e r eA` is a constant, then prove that locus of `z` is a straight line.

z^2+zabsz+absz^2=0 then the locus of z is

if amp (z-1)/(z+1)=pi/3 , then the locus of z is

If abs(z+barz)=abs(z-barz) , then the locus of z is

If (x+y) prop z when y is constant and (z+x) prop y when z is constant . Then prove that (x+y+z) prop yz , when when y and z are both constant.

If w=z/[z-1/(3i)] and |w|=1, then find the locus of z

Given that the complex numbers which satisfy the equation | z bar z ^3|+| bar z z^3|=350 form a rectangle in the Argand plane with the length of its diagonal having an integral number of units, then area of rectangle is 48 sq. units if z_1, z_2, z_3, z_4 are vertices of rectangle, then z_1+z_2+z_3+z_4=0 rectangle is symmetrical about the real axis a r g(z_1-z_3)=pi/4or(3pi)/4

If z!=0 is a complex number, then prove that R e(z)=0 rArr Im(z^2)=0.

If z=r e^(itheta) , then prove that |e^(i z)|=e^(-r sintheta)

If z_1, z_2, z_3 are three nonzero complex numbers such that z_3=(1-lambda)z_1+lambdaz_2 where lambda in R-{0}, then prove that points corresponding to z_1, z_2 and z_3 are collinear .

Identify the locus of z if bar z = bar a +(r^2)/(z-a).