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),zbar_(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)=abs(zbar(z_(0))-1) .

If z_(1) = a + ib " and " z_(2) + c id are complex numbers such that |z_(1)| = |z_(2)| = 1 and Re (z_(1)bar (z)_(2)) = 0 , then the pair of complex numbers w_(1) = a + ic " and " w_(2) = b id satisfies :

If z be a non-zero complex number, show that (bar(z^(-1)))= (bar(z))^(-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_(1) and z_(2) are two complex numbers such that |(z_(1)-z_(2))/(z_(1)+z_(2))|=1 , then

Let z_(1),z_(2) be two complex numbers such that z_(1)+z_(2) and z_(1)z_(2) both are real, then

For two unimobular complex numbers z_(1) and z_(2) , find [(bar(z)_(1),-z_(2)),(bar(z)_(2),z_(1))]^(-1) [(z_(1),z_(2)),(-bar(z)_(2),bar(z)_(1))]^(-1)

If z_1a n dz_2 are two complex numbers such that |z_1|=|z_2|a n d arg(z_1)+a r g(z_2)=pi , then show that z_1,=- bar z _2dot

Number of imaginary complex numbers satisfying the equation, z^2=bar(z)2^(1-|z|) is