If `z+1/z=1` show that `z^4+1/z^4=-1`

if z+(1)/(z)=1 then z^(4)+(1)/(z^(4))=

If z is a complex number such that |z+(1)/(z)|=1 show that Re(z)=0 when |z| is the maximum.

If z=x+iy and |z-1|+|z+1|=4 show that 3x^(2)+4y^(2)=12

if z_(1),z_(2),z_(3),…..z_(n) are complex numbers such that |z_(1)|=|z_(2)| =….=|z_(n)| = |1/z_(1) +1/z_(2) + 1/z_(3) +….+1/z_(n)| =1 Then show that |z_(1) +z_(2) +z_(3) +……+z_(n)|=1

If z=x+iy such that |z+1|=|z-1| and arg((z-1)/(z+1))=(pi)/(4) then

If |z-2|=2|z-1| , then show that |z|^(2)=(4)/(3)Re(z) .

If z=x +i y such that |z+1|=|z-1| and arg((z-1)/(z+1))=pi/4 , then find zdot

If the complex numbers z_1, z_2,.......z_n lie on te unit circle |z| = 1 then show that |z_1 + z_2 + ..... + z_n| = |z_1^(-1)1 + z_2^(-1) + ....... + z_n^(-1)|.

If z_(1),z_(2)inC , show that (z_(1)+z_(2))^(2)=z_(1)^(2)+2z_(1)z_(2)+z_(2)^(2) .

If |z_1|=1,|z_2|=1 then prove that |z_1+z_2|^2+|z_1-z_2|^2 =4.