Question

If $a>0$ and $z|z|+az+2a=0$, then $z$ must be

• A. purely imaginary
• B. a positive real number
• C. a negative real number
• D. having positive imaginary part

Answer 1

The correct option is C a negative real number

$z|z|+az+2a=(x+iy)\sqrt{x^2+y^2}+a(x+iy)+2a=0$
by comparsion
$y\sqrt{x^2+y^2}+ay=0$
y=0,
$x\sqrt{x^2+y^2}+ax+2a=0$
''a'' is positive so x should be negative.
$\therefore z$ is a negative real number.