Let `|z|=2a n dw-(z+1)/(z-1),w h e r ez ,w , in C` (where `C` is the set of complex numbers). Then product of least and greatest value of modulus of `w` is__________.

z and omega are two nonzero complex number such that |z|=|omega|" and "Argz+Arg omega= pi then z equals

If one root of the equation z^2-a z+a-1=0i s(1+i),w h e r ea is a complex number then find the root.

Let z=t^2-1+sqrt(t^4-t^2),w h e r et in R is a parameter. Find the locus of z depending upon t , and draw the lous of z in the Argand plane.

If y z+z x+x y=12 ,w h e r ex ,y ,z are positive values, find the greatest value of x y zdot

If z=x+i ya n dw=(1-i z)//(z-i) , then show that |w|=1 z is purely real.

Let G= {1, w, w^(2)} where w is a complex cube root of unity. Then find the inverse of w^(2) . Under usual multiplication.

If |(z-z_1)/(|z-z_2|)=3,w h e r ez_1a n dz_2 are fixed complex numbers and z is a variable complex number, then z lies on a (a).circle with z_1 as its interior point (b).circle with z_2 as its interior point (c).circle with z_1 as its exterior point (d).circle with z_2 as its exterior point

If z=z_0+A( bar z -( bar z _0)), w h e r eA is a constant, then prove that locus of z is a straight line.

Let z be a complex number satisfying equation z^p=z^(-q),w h e r ep ,q in N ,t h e n (A) if p=q , then number of solutions of equation will be infinite. (B) if p=q , then number of solutions of equation will be finite. (C) if p!=q , then number of solutions of equation will be p+q+1. (D) if p!=q , then number of solutions of equation will be p+qdot

Find the greatest and the least value of |z_1+z_2| if z_1=24+7ia n d|z_2|=6.