Question

Show that $|\frac{z-2}{z-3}|=2$ represents a circle. Find its centre and radius.

Answer 1

Let $z=x+iy$. Then,

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

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

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

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

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

$3x^2+3y^2-20x+32=0$

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

$(x-\frac{10}{3}{)}^2+(y-0{)}^2=(\frac{2}{3}{)}^2$, which represents a circle with centre at $(\frac{10}{3},0)$ and radius $\frac{2}{3}$.