Show that |(z – 2) / (z – 3)| = 2 represents a circle.

According to the question,

We have,

|(z – 2) / (z – 3)| = 2

Substituting z = x + iy, we get

⇒ |(x + iy – 2) / (x + iy – 3)| = 2

⇒ |x – 2 + iy| = 2 |x – 3 + iy|

⇒ √((x – 2)2 + y2) = 2√((x – 3)2 + y2)

⇒ x2 – 4x + 4 + y2 = 4 (x2 – 6x + 9 + y2)

⇒ 3x2 + 3y2 – 20x + 32 = 0

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

$\Rightarrow\left(x-\frac{10}{3}\right)^{2}+y^{2}+\frac{32}{3}-\frac{100}{9}=0$

$\Rightarrow\left(x-\frac{10}{3}\right)^{2}+(y-0)^{2}=\frac{4}{9}$

Therefore, centre of circle is (10/3, 0) and radius is 4/9 or 2/3.