Question

For every real number $c\ge 0,$ find all the complex numbers z which satisfy the equation $2\left|z\right|-4cz+1+ic=0.$

• A. $(x,y)=\frac{4c\pm \sqrt{4c^2+3}}{4(4c^2-1)},1/4$
• B. $(x,y)=\frac{4c\pm \sqrt{4c^2+3}}{4(4c^2+1)},-1/4$
• C. $(x,y)=\frac{4c\pm \sqrt{4c^2+3}}{4(4c^2-1)},-1/4$
• D. $(x,y)=\frac{4c\pm \sqrt{4c^2+3}}{4(4c^2+1)},1/4$

Answer 1

The correct option is A $(x,y)=\frac{4c\pm \sqrt{4c^2+3}}{4(4c^2-1)},1/4$

Let $z=x+iy$
$\Rightarrow 2\sqrt{x^2+y^2}-4c(x+iy)+1+ic=0$
Considering the imaginary part, we get
$\Rightarrow -4cy+c=0$
$\therefore y=\frac{1}{4}$
Substituting in the real part, we get
$\Rightarrow \sqrt{16x^2+1}-8cx+2=0$

$\Rightarrow 16x^2+1=64c^2{x}^2-32cx+4$

$\Rightarrow x^2(64c^2-16)-32cx+3=0$

$\Rightarrow x=\frac{32c\pm \sqrt{1024c^2-768c^2+192}}{2(64c^2-16)}$
$\Rightarrow \frac{8c\pm \sqrt{16c^2+12}}{2(16c^2-4)}$
$\Rightarrow \frac{4c\pm \sqrt{4c^2+3}}{4(4c^2-1)}$