JEE Advance - Mathematics (1999 - No. 17)
For complex numbers z and w, prove that $${\left| z \right|^2}w - {\left| w \right|^2}z = z - w$$ if and only if $$ z = w\,or\,z\overline {\,w} = 1$$.
The statement is true and the provided condition is both necessary and sufficient.
The statement is false; there are complex numbers z and w that satisfy the equation but not the condition.
The condition is necessary, but not sufficient, for the equation to hold.
The condition is sufficient, but not necessary, for the equation to hold.
The equation holds only when z and w are real numbers.