The equation of any Circle in the complex pla | vedclass

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Question

(A) The general equation of a circle in the Cartesian plane is given by $x^2 + y^2 + 2gx + 2fy + c = 0$ ... $(i)$ Let $z = x + iy$ and $\bar{z} = x -

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Circle

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(A) The general equation of a circle in the Cartesian plane is given by $x^2 + y^2 + 2gx + 2fy + c = 0$ ... $(i)$ Let $z = x + iy$ and $\bar{z} = x - iy$. Then $z + \bar{z} = 2x$ and $z \bar{z} = x^2 + y^2$. Also,$y = \frac{z - \bar{z}}{2i} = -\frac{i}{2}(z - \bar{z})$. Substituting these into equation $(i)$: $z \bar{z} + 2g(\frac{z + \bar{z}}{2}) + 2f(\frac{z - \bar{z}}{2i}) + c = 0$ $z \bar{z} + g(z + \bar{z}) - if(z - \bar{z}) + c = 0$ $z \bar{z} + (g - if)z + (g + if)\bar{z} + c = 0$ Let $b = g + if$,then $\bar{b} = g - if$. Substituting these,we get $z \bar{z} + \bar{b}z + b\bar{z} + c = 0$. This represents a circle in the complex plane. Hence,option $A$ is correct.

The equation of any $Circle$ in the complex plane is of the form $z \bar{z} + b \bar{z} + \bar{b} z + c = 0$,where $b \in \mathbb{C}$ and $c \in \mathbb{R}$.

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