Question

Let $z$ be a complex number and $c$ be a real number $\ge$ 1 such that z + $c|z+1|+i=0,$ then $c$ belongs to

• A. $[2,3]$
• B. $(3,4)$
• C. $[1,\sqrt{2}]$
• D. None of these

Answer 1

The correct option is C $[1,\sqrt{2}]$

Since $c$ $|z+1|$ is real
$z+i$ is real
let $z-i=x$, where $x$ is real
$z+i=-c$ $|z+1|$ (given)
$x^2=c^2{\{x+1\}}^2+1^2)$
$(c^2-1)x^2+2c^{2}x+2c^2=0$
As $x$ is real $4c^4-8c^2(c^2-1)\ge 0$
$4c^2-(c^2-2)\le 0$
$c^2\le 2$
$c\le \sqrt{2}$
But $c\ge 1$
$\Rightarrow cϵ[1,\sqrt{2}]$