Numbers of complex numbers z, such that `abs(z)=1` and `abs((z)/bar(z)+bar(z)/(z))=1` is

The number of complex numbers z such that |z-1|=|z+1|=|z-i| is

Let z,z_(0) be two complex numbers. It is given that abs(z)=1 and the numbers z,z_(0),bar(z_(0)),1 and 0 are represented in an Argand diagram by the points P,P_(0),Q,A and the origin, respectively. Show that /_\POP_(0) and /_\AOQ are congruent. Hence, or otherwise, prove that abs(z-z_(0))=abs(zbar(z_(0))-1) .

bb"statement-1" " Let " z_(1),z_(2) " and " z_(3) be htree complex numbers, such that abs(3z_(1)+1)=abs(3z_(2)+1)=abs(3z_(3)+1) " and " 1+z_(1)+z_(2)+z_(3)=0, " then " z_(1),z_(2),z_(3) will represent vertices of an equilateral triangle on the complex plane. bb"statement-2" z_(1),z_(2),z_(3) represent vertices of an triangle, if z_(1)^(2)+z_(2)^(2)+z_(3)^(2)+z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)=0

If z and w are two non-zero complex numbers such that |zw|=1 and Arg (z) -Arg (w) =pi/2 , then bar z w is equal to :

Let z and w be two complex numbers such that |z|le1, |w|le1 and |z-iw|=|z-i bar w|=2 , then z equals :

If z_(1),z_(2) and z_(3), z_(4) are two pairs of conjugate complex numbers, the find the value of arg(z_(1)/z_(4)) + arg(z_(2)//z_(3)) .

If z is any complex number, prove that : |z|^2= |z^2| .

Let z be a non-real complex number lying on |z|=1, prove that z=(1+itan((arg(z))/2))/(1-itan((arg(z))/(2))) (where i=sqrt(-1).)

For any complex number z, prove that : -|z| leR_e (z) le |z| .