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Question Number 19555 by Tinkutara last updated on 12/Aug/17

Find center and radius of circle having  equation zz^�  + (1 − i)z + (1 + i)z^�  − 1 = 0.

$$\mathrm{Find}\:\mathrm{center}\:\mathrm{and}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{circle}\:\mathrm{having} \\ $$$$\mathrm{equation}\:{z}\bar {{z}}\:+\:\left(\mathrm{1}\:−\:{i}\right){z}\:+\:\left(\mathrm{1}\:+\:{i}\right)\bar {{z}}\:−\:\mathrm{1}\:=\:\mathrm{0}. \\ $$

Answered by ajfour last updated on 12/Aug/17

Equation of circle:  ∣z−z_0 ∣=r  ⇒   (z−z_0 )(z^� −z_0 ^� )=r^2   or    zz^� −z_0 ^� z−z_0 z^� +∣z_0 ∣^2 −r^2 =0  comoparing this with given eqn.      z_0 =1+i   and  r=(√(1+∣z_0 ∣^2 )) =(√3) .

$$\mathrm{Equation}\:\mathrm{of}\:\mathrm{circle}: \\ $$$$\mid\mathrm{z}−\mathrm{z}_{\mathrm{0}} \mid=\mathrm{r} \\ $$$$\Rightarrow\:\:\:\left(\mathrm{z}−\mathrm{z}_{\mathrm{0}} \right)\left(\bar {\mathrm{z}}−\bar {\mathrm{z}}_{\mathrm{0}} \right)=\mathrm{r}^{\mathrm{2}} \\ $$$$\mathrm{or}\:\:\:\:\mathrm{z}\bar {\mathrm{z}}−\bar {\mathrm{z}}_{\mathrm{0}} \mathrm{z}−\mathrm{z}_{\mathrm{0}} \bar {\mathrm{z}}+\mid\mathrm{z}_{\mathrm{0}} \mid^{\mathrm{2}} −\mathrm{r}^{\mathrm{2}} =\mathrm{0} \\ $$$$\mathrm{comoparing}\:\mathrm{this}\:\mathrm{with}\:\mathrm{given}\:\mathrm{eqn}. \\ $$$$\:\:\:\:\mathrm{z}_{\mathrm{0}} =\mathrm{1}+\mathrm{i}\:\:\:\mathrm{and}\:\:\mathrm{r}=\sqrt{\mathrm{1}+\mid\mathrm{z}_{\mathrm{0}} \mid^{\mathrm{2}} }\:=\sqrt{\mathrm{3}}\:. \\ $$

Commented by Tinkutara last updated on 12/Aug/17

Thank you very much Sir!

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$

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