Complex Numbers with Graphings Problem


Given that z = w + \frac{1}{w} where w = 2 (\text{cos}\theta + i \text{sin}\theta) = 2 e^{i\theta}, express the real and imaginary parts of z in terms of \theta.

(i) Hence show that the point representing z in an Argand diagram lies on the curve with Cartesian equation \frac{y^2}{9} + \frac{x^2}{25} = \frac{1}{4}.

(ii) Show that |z - 2|^2 = (\frac{5}{2} - 2\text{cos}\theta)^2, and find a similar expression for |z + 2|^2. Deduce that |z - 2| + | z + 2| = 5.

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