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  • Show that |(z-2)/(z-3)|=2 represents a circle. Find its centre and radius.

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

Answers (1)

\left | \frac{z-2}{z-3} \right |=2 Substituting z=x+iy, we get  \left | \frac{x+iy-2}{x+iy-3} \right |=2

|x-2+iy|=2|x-3+iy|

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

x^2-4x+4+y^2=4(x^2-6x+9+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^2+\frac{32}{3}-\frac{100}{9}=0

Thus, the centre of circle is (\frac{10}{3},0) and radius is 2/3

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