Intersection of two loci

7

1. Without using a calculator, solve the equation $$3 w + 2 \mathrm { i } w ^ { * } = 17 + 8 \mathrm { i }$$ where \(w ^ { * }\) denotes the complex conjugate of \(w\). Give your answer in the form \(a + b \mathrm { i }\).
2. In an Argand diagram, the loci $$\arg ( z - 2 \mathrm { i } ) = \frac { 1 } { 6 } \pi \quad \text { and } \quad | z - 3 | = | z - 3 \mathrm { i } |$$ intersect at the point \(P\). Express the complex number represented by \(P\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), giving the exact value of \(\theta\) and the value of \(r\) correct to 3 significant figures.

8 The complex number 1 - i is denoted by \(u\).

1. Showing your working and without using a calculator, express $$\frac { \mathrm { i } } { u }$$ in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. On an Argand diagram, sketch the loci representing complex numbers \(z\) satisfying the equations \(| z - u | = | z |\) and \(| z - \mathrm { i } | = 2\).
3. Find the argument of each of the complex numbers represented by the points of intersection of the two loci in part (ii).

7

1. On a single Argand diagram sketch the loci given by the equations \(| z - 3 + 2 i | = 2\) and \(| w - 3 + 2 \mathrm { i } | = | w + 3 - 4 \mathrm { i } |\) where z and \(w\) are complex numbers.
2. Hence find the least value of \(| \mathbf { z } - \mathbf { w } |\) for points on these loci. Give your answer in an exact form.

7. The point \(P\) represents a complex number \(z\) on an Argand diagram, where $$| z + 1 | = | 2 z - 1 |$$ and the point \(Q\) represents a complex number \(w\) on the Argand diagram, where $$| w | = | w - 1 + \mathrm { i } |$$ Find the exact coordinates of the points where the locus of \(P\) intersects the locus of \(Q\).

1. The point \(P\) represents a complex number \(z\) on an Argand diagram such that

$$| z - 6 \mathrm { i } | = 2 | z - 3 |$$

1. Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle. The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$\arg ( z - 6 ) = - \frac { 3 \pi } { 4 }$$
2. Sketch, on the same Argand diagram, the locus of \(P\) and the locus of \(Q\) as \(z\) varies.
3. Find the complex number for which both \(| z - 6 \mathrm { i } | = 2 | z - 3 |\) and \(\arg ( z - 6 ) = - \frac { 3 \pi } { 4 }\)

6. The complex number \(z\) on an Argand diagram is represented by the point \(P\) where $$| z + 1 - 13 i | = 3 | z - 7 - 5 i |$$ Given that the locus of \(P\) is a circle,

1. determine the centre and radius of this circle. The complex number \(w\), on the same Argand diagram, is represented by the point \(Q\), where $$\arg ( w - 8 - 6 \mathrm { i } ) = - \frac { 3 \pi } { 4 }$$ Given that the locus of \(P\) intersects the locus of \(Q\) at the point \(R\),
2. determine the complex number representing \(R\).

6 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by $$| z | = | z - 4 \mathbf { i } | \quad \text { and } \quad \arg z = \frac { 1 } { 6 } \pi$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Hence find, in the form \(x +\) i \(y\), the complex number represented by the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).

6 In an Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by $$| z | = 2 \quad \text { and } \quad \arg z = \frac { 1 } { 3 } \pi$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Hence find, in the form \(x + \mathrm { i } y\), the complex number representing the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).

5

1. Sketch the locus \(| z - ( 3 + 4 j ) | = 2\) on an Argand diagram.
2. On the same diagram, sketch the locus \(\arg ( z - 4 ) = \frac { 1 } { 2 } \pi\).
3. Indicate clearly on your sketch the points which satisfy both $$| z - ( 3 + 4 j ) | = 2 \quad \text { and } \quad \arg ( z - 4 ) = \frac { 1 } { 2 } \pi$$

2 Indicate on a single Argand diagram

1. the set of points for which \(| z - ( - 3 + 2 \mathrm { j } ) | = 2\),
2. the set of points for which \(\arg ( z - 2 \mathrm { j } ) = \pi\),
3. the two points for which \(| z - ( - 3 + 2 \mathrm { j } ) | = 2\) and \(\arg ( z - 2 \mathrm { j } ) = \pi\).

6 Sketch, on a single Argand diagram, the loci given by \(| z - \sqrt { 3 } - \mathrm { i } | = 2\) and \(\arg z = \frac { 1 } { 6 } \pi\).

5 The complex number \(1 + \mathrm { i } \sqrt { 3 }\) is denoted by \(a\).

1. Find \(| a |\) and \(\arg a\).
2. Sketch on a single Argand diagram the loci given by \(| z - a | = | a |\) and \(\arg ( z - a ) = \frac { 1 } { 2 } \pi\).

7 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - 3 - 4 \mathrm { i } | = 4\) and \(| z | = | z - 8 \mathrm { i } |\) respectively.

1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Hence find the complex numbers represented by the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
3. Indicate, by shading, the region of the Argand diagram for which $$| z - 3 - 4 i | \leqslant 4 \text { and } | z | \geqslant | z - 8 i | .$$

5 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z + 2 | = 2\) and \(\arg ( z + 2 ) = \frac { 5 } { 6 } \pi\) respectively.

1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Find the complex number represented by the intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
3. Indicate, by shading, the region of the Argand diagram for which $$| z + 2 | \leqslant 2 \text { and } \frac { 5 } { 6 } \pi \leqslant \arg ( z + 2 ) \leqslant \pi .$$

7 In this question you must show detailed reasoning.
Two loci, \(C _ { 1 }\) and \(C _ { 2 }\), are defined as follows. \(\mathrm { C } _ { 1 } = \left\{ \mathrm { z } : \arg ( \mathrm { z } + 2 - \mathrm { i } ) = \frac { 1 } { 4 } \pi \right\}\) and \(\mathrm { C } _ { 2 } = \left\{ \mathrm { z } : \arg ( \mathrm { z } - 2 - \sqrt { 3 } - 2 \mathrm { i } ) = \frac { 2 } { 3 } \pi \right\}\) By considering the representations of \(C _ { 1 }\) and \(C _ { 2 }\) on an Argand diagram, determine the locus \(C _ { 1 } \cap C _ { 2 }\).

4 The Argand diagram shows a circle of radius 3. The centre of the circle is the point which represents the complex number \(4 - 2 \mathrm { i }\). \includegraphics[max width=\textwidth, alt={}, center]{4159328b-475e-4f29-91f2-f2f343573251-3_417_775_349_644}

1. 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 \(\mathrm { L } = \{ \mathrm { z } : \mathrm { z } \in \mathbb { C } , | \mathrm { z } - \mathrm { i } | = | \mathrm { z } + 2 | \}\).
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.
3. Find this number.

2

1. On the same Argand diagram, draw:
   1. the locus of points satisfying \(| z - 4 + 2 \mathrm { i } | = 4\);
   2. the locus of points satisfying \(| z | = | z - 2 \mathrm { i } |\).
2. Indicate on your sketch the set of points satisfying both $$| z - 4 + 2 i | \leqslant 4$$ and $$| z | \geqslant | z - 2 \mathrm { i } |$$

4

1. A circle \(C\) in the Argand diagram has equation $$| z + 5 - \mathrm { i } | = \sqrt { 2 }$$ Write down its radius and the complex number representing its centre.
2. A half-line \(L\) in the Argand diagram has equation $$\arg ( z + 2 \mathrm { i } ) = \frac { 3 \pi } { 4 }$$ Show that \(z _ { 1 } = - 4 + 2 \mathrm { i }\) lies on \(L\).
3. 1. Show that \(z _ { 1 } = - 4 + 2 \mathrm { i }\) also lies on \(C\).
   2. Hence show that \(L\) touches \(C\).
   3. Sketch \(L\) and \(C\) on one Argand diagram.
4. The complex number \(z _ { 2 }\) lies on \(C\) and is such that \(\arg \left( z _ { 2 } + 2 \mathrm { i } \right)\) has as great a value as possible. Indicate the position of \(z _ { 2 }\) on your sketch.

3 Two loci, \(L _ { 1 }\) and \(L _ { 2 }\), in an Argand diagram are given by $$\begin{aligned} & L _ { 1 } : | z + 1 + 3 \mathrm { i } | = | z - 5 - 7 \mathrm { i } | \\ & L _ { 2 } : \arg z = \frac { \pi } { 4 } \end{aligned}$$

1. Verify that the point represented by the complex number \(2 + 2 \mathrm { i }\) is a point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).
2. Sketch \(L _ { 1 }\) and \(L _ { 2 }\) on one Argand diagram.
3. Shade on your Argand diagram the region satisfying
   both $$| z + 1 + 3 i | \leqslant | z - 5 - 7 i |$$ and $$\frac { \pi } { 4 } \leqslant \arg z \leqslant \frac { \pi } { 2 }$$

2

1. Draw on the Argand diagram below:
   1. the locus of points for which $$| z - 2 - 3 \mathrm { i } | = 2$$
   2. the locus of points for which $$| z + 2 - \mathrm { i } | = | z - 2 |$$
2. Indicate on your diagram the points satisfying both $$| z - 2 - 3 \mathrm { i } | = 2$$ and $$| z + 2 - \mathrm { i } | \leqslant | z - 2 |$$ (l mark) \includegraphics[max width=\textwidth, alt={}, center]{ff63460d-0fa1-437d-bc08-3e7ce809e32b-3_1404_1431_1043_319}

7

1. Sketch on a single Argand diagram
   1. the set of points for which \(| z - 1 - 3 i | = 3\),
   2. the set of points for which \(\arg ( z + 4 ) = \frac { 1 } { 4 } \pi\).
2. Find, in exact form, the two values of \(z\) for which \(| z - 1 - 3 i | = 3\) and \(\arg ( z + 4 ) = \frac { 1 } { 4 } \pi\).

8 In an Argand diagram, the point P representing the complex number \(w\) lies on the locus defined by \(\left\{ z : \arg ( z - 7 ) = \frac { 3 } { 4 } \pi \right\}\). You are given that \(\operatorname { Re } ( w ) = 1\).

1. Find \(w\). The point P also lies on the locus defined by \(\{ \mathrm { z } : | \mathrm { z } + 3 - 9 \mathrm { i } | = \mathrm { k } \}\), where \(k\) is a constant.
2. Find the complex number represented by the other point of intersection of the loci defined by $$\{ z : | z + 3 - 9 i | = k \} \text { and } \left\{ z : \arg ( z - 7 ) = \frac { 3 } { 4 } \pi \right\} .$$

9

1. On a single Argand diagram, sketch the loci defined by
   The point of intersection of the two loci in part (a) represents the complex number \(w\). Find \(w\), giving your answer in exact form. \section*{END OF QUESTION PAPER}

8 Two sets of complex numbers are given by \(\left\{ z : \arg ( z - 10 ) = \frac { 3 } { 4 } \pi \right\}\) and \(\{ z : | z - 3 - 6 i | = k \}\), where \(k\) is a positive constant. In an Argand diagram, one of the points of intersection of the two loci representing these sets lies on the imaginary axis.

1. Sketch the loci on an Argand diagram.
2. In this question you must show detailed reasoning. Find the complex numbers represented by the points of intersection.

13

1. On separate Argand diagrams, show the set of points representing each of the following inequalities.
   1. \(| z | \leqslant \sqrt { 5 }\)
   2. \(\quad | z + 2 - 4 i | \geqslant | z - 2 - 6 i |\)
2. Show that there is a unique value of \(z\), which should be determined, for which both \(| z | \leqslant \sqrt { 5 }\) and \(| z + 2 - 4 i | \geqslant | z - 2 - 6 i |\).