Complex Numbers with Graphings Problem

N95/II/12

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 $|z - 2| + | z + 2| = 5$ .

Solution

About

KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for the more than 10 years. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

Complex Numbers with Graphings Problem

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