Question

$\left|\frac{z-2}{z-3}\right|=2$ represents a circle, find its radius.

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $3$
• D. $\frac{3}{2}$

Answer 1

The correct option is B $\frac{2}{3}$

Let $z=x+iy$. Then,

$|\frac{z-2}{z-3}|=2$

$|\frac{(x-2)+iy}{(x-3)+iy}|=2$

$|(x-2)+iy|=2|(x-3)+iy|$

$\sqrt{(x-2{)}^2+y^2}=2\sqrt{(x-3{)}^2+y^2}$

$(x-2{)}^2+y^2=4[(x-3{)}^2+y^2]$

$x^2+y^2-\frac{20}{3}x+\frac{32}{3}=0$

$x^2-\frac{20}{3}x+y^2+\frac{32}{3}+\frac{100}{9}-\frac{100}{9}=0$

$(x-\frac{10}{3}{)}^2+(y-0{)}^2=(\frac{2}{3}{)}^2$

Hence, radius = $\frac{2}{3}$