Question

If $z=x+iy$ is a complex number satisfying $\left|\frac{z-2i}{z+2i}\right|=2$ and the locus of $z$ is a circle, then its radius is

• A. $\frac{5}{3}$
• B. $\sqrt{\frac{71}{9}}$
• C. $\frac{8}{3}$
• D. $\frac{1}{3}$

Answer 1

Answer: (C)

We have,
$\left|\frac{z-2i}{z+2i}\right|=2$
$\Rightarrow \left|\frac{x+iy-2i}{x+iy+2i}\right|=2$
$\Rightarrow \left|\frac{x+(y-2)i}{x+(y+2)i}\right|=2$
$\Rightarrow \frac{|x+(y-2)i|}{|x+(y+2)i|}=2$
$\Rightarrow \frac{\sqrt{x^2+(y-2{)}^2}}{\sqrt{x^2+(y+2{)}^2}}=2$
$\Rightarrow x^2+(y-2{)}^2=4\left[x^2+(y+2{)}^2\right]$
$\Rightarrow x^2+y^2-4y+4=4x^2+4y^2+16y+16$
$\Rightarrow 3x^2+3y^2+20y+12=0$
$\Rightarrow x^2+y^2+\frac{20}{3}y+4=0$
$\Rightarrow x^2+{\left(y+\frac{10}{3}\right)}^2+4-\frac{100}{9}=0$
$\Rightarrow x^2+{\left(y+\frac{10}{3}\right)}^2={\left(\frac{8}{3}\right)}^2$
Which is equation of circle with centre $\left(0,\frac{-10}{3}\right)$ and radius $\frac{8}{3}$.