Explain the falacy:

Find the circle roots of -27.

Find the principal value of -2i.

Show that \(\left| \frac { z-3 }{ z+3 } \right| \) = 2 represent a circle.

Show that the complex numbers 3 + 2i, 5i, -3 + 2i and -i form a square.

If the complex number 2 + i and 1-2i are equidistant from x + iy then show that x+3y = 0.

Find the locus of z if Re\(\left( \frac { z+1 }{ z-i } \right) \) = 0 where z = x+iy.

Find the locus of z if |3z - 5| = 3 |z + 1| where z = x + iy.

Find the locus of z if Re\(\\ \left( \frac { \bar { z } +1 }{ \bar { z } -i } \right) \) = 0.

If \(\frac { (a+i)^{ 2 } }{ 2a-i } \) = p + iq, show that p²+q² = \(\frac { ({ a }^{ 2 }+i)^{ 2 } }{ 4a^{ 2 }+1 } \).