For complex numbers `z and w`, prove that `|z|^2w- |w|^2 z = z - w`, if and only if `z = w` or `z barw = 1`.

For any two complex numbers z_1 and z_2 , prove that |z_1+z_2| =|z_1|-|z_2| and |z_1-z_2|>=|z_1|-|z_2|

Let za n dw be two nonzero complex numbers such that |z|=|w|a n d arg(z)+a r g(w)=pidot Then prove that z=- bar w dot

It z_(1) and z_(2) are two complex numbers, such that |z_(1)| = |z_(2)| , then is it necessary that z_(1) = z_(2) ?

Evaluate the following if z = 5 -2i and w = -1 + 3i z + w

Evaluate the following if z = 5 -2i and w = -1 + 3i 2z + 3w

Let z and omega be two complex numbers such that |z|lt=1,|omega|lt=1 and |z-iomega|=|z-i bar omega|=2, then z equals (a) 1ori (b). ior-i (c). 1or-1 (d). ior-1

Evaluate the following if z = 5 -2i and w = -1 + 3i z - iw

If Pa n dQ are represented by the complex numbers z_1a n dz_2 such that |1/z_2+1/z_1|=|1/(z_2)-1/z_1| , then O P Q(w h e r eO) is the origin is equilateral O P Q is right angled. the circumcenter of O P Qi s1/2(z_1+z_2) the circumcenter of O P Qi s1/3(z_1+z_2)

For any two complex numbers z_(1)" and "z_(2) prove that Im(z_(1)z_(2))=Re(z_1) Im(z_2)+ Im(z_1)Re(z_2) .