Locus with parameter variation - Complex Numbers Argand & Loci

CAIE P3 2008 June Q5

7 marks Standard +0.8

5 The variable complex number $z$ is given by $$z = 2 \cos \theta + \mathrm { i } ( 1 - 2 \sin \theta ) ,$$ where $\theta$ takes all values in the interval $- \pi < \theta \leqslant \pi$.

Show that $| z - \mathrm { i } | = 2$, for all values of $\theta$. Hence sketch, in an Argand diagram, the locus of the point representing $z$.
Prove that the real part of $\frac { 1 } { z + 2 - \mathrm { i } }$ is constant for $- \pi < \theta < \pi$.

CAIE P3 2024 November Q8

9 marks Standard +0.3

8

Given that $z = 1 + y \mathrm { i }$ and that $y$ is a real number, express $\frac { 1 } { z }$ in the form $a + b \mathrm { i }$, where $a$ and $b$ are functions of $y$.
Show that $\left( a - \frac { 1 } { 2 } \right) ^ { 2 } + b ^ { 2 } = \frac { 1 } { 4 }$, where $a$ and $b$ are the functions of $y$ found in part (a). \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-14_2716_35_108_2012}
On a single Argand diagram, sketch the loci given by the equations $\operatorname { Re } ( z ) = 1$ and $\left| z - \frac { 1 } { 2 } \right| = \frac { 1 } { 2 }$, where $z$ is a complex number.
The complex number $z$ is such that $\operatorname { Re } ( z ) = 1$. Use your answer to part (b) to give a geometrical description of the locus of $\frac { 1 } { z }$.

OCR FP3 2011 June Q2

6 marks Standard +0.8

It is given that $z = e^{i\theta}$, where $0 < \theta < 2\pi$, and $w = \frac{1+z}{1-z}$.

Prove that $w = i \cot \frac{1}{2}\theta$. [3]
Sketch separate Argand diagrams to show the locus of $z$ and the locus of $w$. You should show the direction in which each locus is described when $\theta$ increases in the interval $0 < \theta < 2\pi$. [3]