4 The Argand diagram shows a circle of radius 3. The centre of the circle is the point which represents the complex number \(4 - 2 \mathrm { i }\).
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- Use set notation to define the locus of complex numbers, \(z\), represented by points which lie on the circle.
The locus \(L\) is defined by \(\mathrm { L } = \{ \mathrm { z } : \mathrm { z } \in \mathbb { C } , | \mathrm { z } - \mathrm { i } | = | \mathrm { z } + 2 | \}\).
- On the Argand diagram in the Printed Answer Booklet, sketch and label the locus \(L\).
You are given that the locus \(\left\{ z : z \in \mathbb { C } , \arg ( z - 1 ) = \frac { 1 } { 4 } \pi , \operatorname { Re } ( z ) = 3 \right\}\) contains only one number.
- Find this number.