Question

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

Answer 1

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

$\Rightarrow \left|\frac{n+iy-2}{n+iy-3}\right|=2\Rightarrow \left|\frac{(x-2)+iy}{(x-3)+iy}\right|=2$

$\frac{\sqrt{(x-2{)}^2+y^2}}{\sqrt{(x-3{)}^2+y^2}}=2\Rightarrow \frac{(x-2{)}^2+y^2}{(x-3{)}^2+y^2}=4$

$x^2+4-4x+y^2=4x^2-24x+36+4y^2$

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

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

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

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

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

$\therefore$Centre = $(\frac{10}{3},0)$

$\therefore \Rightarrow r=\pm \frac{2}{3}.$

So radius $=\frac{2}{3}$