Show that for ` z epsilon C, |\z|=0 ` if and only if ` z=0`

Define p on z by for m,nin z, mpn if and only if mn>=0. Check whether p is an equivalence relation on z.

Let I Arg((z-8i)/(z+6))=pmpi/2 II: Re ((z-8i)/(z+6))=0 Show that locus of z in I or II lies on x^2+y^2+6^x – 8^y=0 Hence show that locus of z can also be represented by (z-8i)/(z+6)+(bar(z)-8i)/(barz+6)=0 Further if locus of z is expressed as |z + 3 – 4i| = R, then find R.

Show that |a b c a+2x b+2y c+2z x y z|=0

Show that if iz^(3)+z^(2)-z+i=0, then |z|=1

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 If a , b gt 0 and c lt 0 , then

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.

Prove that the complex numbers z_(1),z_(2) and the origin form an equilateral triangle only if z_(1)^(2) + z_(2)^(2) - z_(1)z_(2)=0 .

for a complex number z, Show that |z|=0 z=0