Complex Numbers

If \({\left( {{{1 + i} \over {1 - i}}} \right)^x} = 1\) then :

If \(z\) and \(\omega\) are two non-zero complex numbers such that \(\left| {z\omega } \right| = 1\) and \(Arg(z) - Arg(\omega ) = {\pi \over 2},\) then \(\,\overline {z\,} \omega\) is equal to

Let \({Z_1}\) and \({Z_2}\) be two roots of the equation \({Z^2} + aZ + b = 0\), Z being complex. Further , assume that the origin, \({Z_1}\) and \({Z_2}\) form an equilateral triangle. Then :

The locus of the centre of a circle which touches the circle \(\left| {z - {z_1}} \right| = a\) and\(\left| {z - {z_2}} \right| = b\,\) externally
(\(z,\,{z_1}\,\& \,{z_2}\,\) are complex numbers) will be :

If \(\left| {z - 4} \right| < \left| {z - 2} \right|\), its solution is given by :

z and w are two nonzero complex numbers such that \(\,\left| z \right| = \left| w \right|\) and Arg z + Arg w =\(\pi\) then z equals