If `z=a+ib` and `Z=A+iB` then show thatif `z=(i(Z+1))/(z-1)` then `a^2+b^2-a=((A^2+B^2+2A-2B+1)/((A-1)^2+B^2))`

If z=a+ib and |z-2|=|2z-1|prove that a^2-b^2=1

If x,y,b and real, z=x+iy and (z-i)/(z-1)=ib, show that, (x-(1)/(2))^(2)+(y-(1)/(2))^(2)=(1)/(2).

Write the following in z=a+ib form: z=1/(3-2i)

z_1, z_2, z_3 are three non-zero complex numbers such that z_2 ne z_1, and a = |z_1|, b = |z_2|, c= |z_3|. If abs{:(a,b, c),(b, c, a),(c, a, b):}=0 , then show that arg. (z_3)/(z_2)="arg" ((z_3-z_1)/(z_2 - z_1))^2 .

If z = (2-i)/(i) , then R e(z^(2)) + I m(z^(2)) is equal to a)1 b)-1 c)2 d)-2

If z_(1)=a+ib and z_(2)=c+id are complex numbers such that Re(z_(1)bar(z)_(2))=0, then the |z_(1)|=|z_(2)|=1 and Re(z_(1)bar(z)_(2))=0, then the pair ofcomplex nunmbers omega=a+ic and omega_(2)=b+id satisfies

Read the following writeup carefully: If z_1 = a+ib and z_2 =c + id be two complex numbers such that |z_1| = |z_2|=1 and "Re" (z_1 bar(z_2))=0 . Now answer the following question Let k = |z_1 + z_2| + |a+ ib| , then the value of k is