If z=a+ib, |z|=1 and b`ne`0, show that z can be represented as `z=(c+i)/(c-i)` where c is a real number.

show that if |(z-3i)/(z+3i)|=1 then z is a real number.

If |(z-5i)/(z+5i)|=1 show that z is a real number

If |z + i| = |z - i|, prove that z is real.

If z ne 1 and (z^(2))/(z-1) is real, the point represented by the complex numbers z lies

If z=(2-i)/(1-i) the real part of z+(1)/(z)=?

If z in C then |(2z-i)/(5z+1)|=m,m in R represents a straight line if

Let z=a+ib=re^(i theta) where a, b, theta in R and i=sqrt(-1) Then r=sqrt((a^(2)+b^(2)))=|Z| and theta=tan^(-1)((b)/(a))=arg(z) Now |z|^(2)=a^(2)+b^(2)=(a+ib)(a-ib)=zbar(z) rArr(1)/(2)=(bar(z))/(|z|^(2)) and |z_(1)z_(2)z_(3)......z_(n)|=|z_(1)||z_(2)||z_(3)|...|z_(n)| If |f(z)|=1 ,then f(z) is called unimodular. In this case f(z) can always be expressed as f(z)=e^(i alpha), alpha in R Also e^(i alpha)+e^(i beta)=e^(i((alpha+beta)/(2)))*2cos((alpha-beta)/(2)) and e^(i alpha)-e^(i beta)=e^(i((alpha+beta)/(2)))*2i sin((alpha-beta)/(2)) where alpha, beta in R Q:If Z_(1),Z_(2),Z_(3) are complex number such that |Z_(1)|=|Z_(2)|=|Z_(3)|=|Z_(1)+Z_(2)+Z_(3)|=1 , then |(1)/(Z_(1))+(1)/(Z_(2))+(1)/(Z_(3))| is

If z=x+iy, where i=sqrt(-1) , then the equation abs(((2z-i)/(z+i)))=m represents a circle, then m can be

If z in C and 2z = |z| + i , then z equals

Show that for z varepsilon C,|z|=0 if and only if z=0