Algebra and Geometry | Part IA, 2004

Simplify the fraction


where zˉ\bar{z} is the complex conjugate of zz. Determine the geometric form that satisfies


Find solutions to

Im(logz)=π3\operatorname{Im}(\log z)=\frac{\pi}{3}


z2=x2y2+2ix,z^{2}=x^{2}-y^{2}+2 i x,

where z=x+iyz=x+i y is a complex variable. Sketch these solutions in the complex plane and describe the region they enclose. Derive complex equations for the circumscribed and inscribed circles for the region. [The circumscribed circle is the circle that passes through the vertices of the region and the inscribed circle is the largest circle that fits within the region.]

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