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If the ratio (z-i)/(z-1) is purely imagi...

If the ratio `(z-i)/(z-1)` is purely imaginary, prove that the point z lies on the circle whose centre is the point `(1)/(2) (1+i)` and radius is `(1)/(sqrt2)`

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we have , `(z-i)/(z-1)` is purely imaginary
`Rightarrow (z-i)/(z -1)+ (barz+i)/(barz-1)= 0`
` Rightarrow zbarz-z-bar(iz) + i+zbarz +iz -barz-i =0`
`Rightarrow 2zbarz+ ( -1 +i) z(-1-i)barz=0`
` Rightarrow zbarz - ((1-i)/2) z- ((1 + i)/2) barz =0`
which represents a circle centred at `((1+i)/2)` and radius ` = sqrt(alpha baralpha-c) = sqrt(((1+i))/2((1-i))/2) = 1/sqrt2`
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