Question

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

• A. $\frac{2}{3}$
• B. $\frac{4}{3}$
• C. $\frac{2}{\sqrt{3}}$
• D. $\frac{4}{\sqrt{3}}$

Answer 1

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

$\left|\frac{z-2}{z-3}\right|=2$
or ${|z-2|}^2=4{|z-3|}^2$
$⟹{|x-2+iy|}^2=4{|x-3+iy|}^2$
or $(x-2{)}^2+y^2=4[(x-3{)}^2+y^2]$
or $3x^2+3y^2-24x+4x+36-4=0$
or $x^2+y^2-\frac{20}{3}x+\frac{32}{3}=0$
This represents a circle with center $(\frac{10}{3},0)$ and radius $\sqrt{\left(\frac{100}{9}\right)-\left(\frac{32}{3}\right)}=\sqrt{\frac{4}{9}}=\frac{2}{3}$