If the point $z$ in the complex plane describes a circle with radius $2$ with centre at the origin then the point $z+\frac{1}{z}$ describe...


A Circle

A parabola

an ellipse

a hyperbola

What I did... Did some elementary manipulations on the given term and conditions but couldn't really equate my manipulations to anything.

  • $\begingroup$ Can you explain why? $\endgroup$
    – darthsid
    May 10, 2017 at 4:23

1 Answer 1


Putting $z=2e^{it}$ gives the $x$ and $y$ coordinates of the transformed shape as $x=(5/2)\cos t$ and $y=(3/2)\sin t$. These parametrise the ellipse $$\frac{x^2}{(5/2)^2}+\frac{y^2}{(3/2)^2}=1.$$


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