An algebra problem by Snehashis Mukherjee

The circular point $z$ is described by $z = 2e^{i\theta}.$ If we are interested in the curve described by $z + \frac{1}{z}$ , then:

$z + \frac{1}{z} = 2e^{i\theta} + \frac{1}{2} e^{-i\theta} = 2(cos(\theta) + i \cdot sin(\theta)) + \frac{1}{2}(cos(\theta) - i \cdot sin(\theta)) = \boxed{\frac{5}{2} cos(\theta) + i \cdot \frac{3}{2} sin(\theta)}$

which is an ellipse.