4.02o Loci in Argand diagram: circles, half-lines

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 $$| z - 3 - 4 i | \leqslant 4 \text { and } | z | \geqslant | z - 8 i | .$$

7 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(\arg ( z - 2 - 2 \mathrm { i } ) = \frac { 1 } { 4 } \pi\) and \(| z | = | z - 10 |\) respectively.

1. Sketch on a single Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Indicate, by shading, the region of the Argand diagram for which $$0 \leqslant \arg ( z - 2 - 2 \mathrm { i } ) \leqslant \frac { 1 } { 4 } \pi \text { and } | z | \geqslant | z - 10 | .$$

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 $$| z + 2 | \leqslant 2 \text { and } \frac { 5 } { 6 } \pi \leqslant \arg ( z + 2 ) \leqslant \pi .$$

4 Write down the equation of the locus represented in the Argand diagram shown in Fig. 4. \begin{figure}[h] \end{figure}

8

1. Fig. 8 shows an Argand diagram. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{df275813-15de-496f-9742-427a9e03f431-3_892_899_1048_664} \captionsetup{labelformat=empty} \caption{Fig. 8}
   \end{figure}
   1. Write down the equation of the locus represented by the circumference of circle B.
   2. Write down the two inequalities that define the shaded region between, but not including, circles A and B.
2. 1. Draw an Argand diagram to show the region where $$\frac { \pi } { 4 } < \arg ( z - ( 2 + \mathrm { j } ) ) < \frac { 3 \pi } { 4 }$$
   2. Determine whether the point \(43 + 47 \mathrm { j }\) lies within this region.

8

1. Sketch on an Argand diagram the locus, \(C\), of points for which \(| z - 4 | = 3\).
2. By drawing appropriate lines through the origin, indicate on your Argand diagram the point A on the locus \(C\) where \(\arg z\) has its maximum value. Indicate also the point B on the locus \(C\) where \(\arg z\) has its minimum value.
3. Given that \(\arg z = \alpha\) at A and \(\arg z = \beta\) at B , indicate on your Argand diagram the set of points for which \(\beta \leqslant \arg z \leqslant \alpha\) and \(| z - 4 | \geqslant 3\).
4. Calculate the value of \(\alpha\) and the value of \(\beta\).

8

1. Indicate on an Argand diagram the set of points \(z\) for which \(| z - ( - 8 + 15 \mathrm { j } ) | < 10\).
2. Using the diagram, show that \(7 < | z | < 27\).
3. Mark on your Argand diagram the point, \(P\), at which \(| z - ( - 8 + 15 \mathrm { j } ) | = 10\) and \(\arg z\) takes its maximum value. Find the modulus and argument of \(z\) at \(P\).

8 Fig. 8 shows an Argand diagram. \begin{figure}[h] \end{figure}

1. Write down the equation of the locus represented by the perimeter of the circle in the Argand diagram.
2. Write down the equation of the locus represented by the half-line \(\ell\) in the Argand diagram.
3. Express the complex number represented by the point P in the form \(a + b \mathrm { j }\), giving the exact values of \(a\) and \(b\).
4. Use inequalities to describe the set of points that fall within the shaded region (excluding its boundaries) in the Argand diagram.

8 You are given the complex number \(w = 2 + 2 \sqrt { 3 } \mathrm { j }\).

1. Express \(w\) in modulus-argument form.
2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$- \frac { \pi } { 2 } \leqslant \arg z \leqslant \frac { \pi } { 3 } \text { and } | z | \leqslant 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(| z - w |\).

4 Indicate, on a single Argand diagram

1. the set of points for which \(\arg ( z - ( - 1 - \mathrm { j } ) ) = \frac { \pi } { 4 }\),
2. the set of points for which \(| z - ( 1 + 2 j ) | = 2\),
3. the set of points for which \(| z - ( 1 + 2 j ) | \geqslant 2\) and \(0 \leqslant \arg ( z - ( - 1 - j ) ) \leqslant \frac { \pi } { 4 }\).

5 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z + 3 - 4 \mathrm { j } | = 5\) and arg \(( z + 3 - 6 \mathrm { j } ) = \frac { 1 } { 2 } \pi\) respectively.

1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Write down the complex number represented by the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
3. Indicate, by shading on your sketch, the region satisfying $$| z + 3 - 4 \mathrm { j } | \geqslant 5 \text { and } \frac { 1 } { 2 } \pi \leqslant \arg ( z + 3 - 6 \mathrm { j } ) \leqslant \frac { 3 } { 4 } \pi .$$

8

1. Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Show that \(z - \frac { 1 } { z } = 2 \mathrm { i } \sin \theta\) and hence express \(16 \sin ^ { 5 } \theta\) in the form \(\sin 5 \theta + p \sin 3 \theta + q \sin \theta\), where \(p\) and \(q\) are integers to be determined.
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } 16 \sin ^ { 5 } \theta \mathrm {~d} \theta\).

3 In this question you must show detailed reasoning.
The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by \(z _ { 1 } = 2 - 3 i\) and \(z _ { 2 } = a + 4 i\) where \(a\) is a real number.

1. Express \(z _ { 1 }\) in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures.
2. Find \(z _ { 1 } z _ { 2 }\) in terms of \(a\), writing your answer in the form \(c + \mathrm { id }\).
3. The real and imaginary parts of a complex number on an Argand diagram are \(x\) and \(y\) respectively. Given that the point representing \(z _ { 1 } z _ { 2 }\) lies on the line \(y = x\), find the value of \(a\).
4. Given instead that \(z _ { 1 } z _ { 2 } = \left( z _ { 1 } z _ { 2 } \right) ^ { * }\) find the value of \(a\).

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.

4 Draw the region of the Argand diagram for which \(| z - 3 - 4 i | \leq 5\) and \(| z | \leq | z - 2 |\).

2 Indicate by shading on an Argand diagram the region \(\{ z : | z | \leqslant | z - 4 | \} \cap \{ z : | z - 3 - 2 i | \leqslant 2 \}\).

3 In this question you must show detailed reasoning.

1. Find the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\). The loci \(\mathrm { C } _ { 1 }\) and \(\mathrm { C } _ { 2 }\) are given by \(| z | = | z - 2 \mathrm { i } |\) and \(| z - 2 | = \sqrt { 5 }\) respectively.
2. 1. Sketch on a single Argand diagram the loci \(\mathrm { C } _ { 1 }\) and \(\mathrm { C } _ { 2 }\), showing any intercepts with the imaginary axis.
   2. Indicate, by shading on your Argand diagram, the region $$\{ z : | z | \leqslant | z - 2 i | \} \cap \{ z : | z - 2 | \leqslant \sqrt { 5 } \} .$$
3. 1. Show that both of the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\) satisfy \(| z - 2 | < \sqrt { 5 }\).
   2. State, with a reason, which root of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\) satisfies \(| z | < | z - 2 i |\).
4. On the same Argand diagram as part (b), indicate the positions of the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\).

2 The locus \(C _ { 1 }\) is defined by \(C _ { 1 } = \left\{ z : 0 \leqslant \arg ( z + i ) \leqslant \frac { 1 } { 4 } \pi \right\}\).

1. Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing \(C _ { 1 }\).
2. Determine whether the complex number \(1.2 + 0.8\) is is \(C _ { 1 }\). The locus \(C _ { 2 }\) is the set of complex numbers represented by the interior of the circle with radius 2 and centre 3 . The locus \(C _ { 2 }\) is illustrated on the Argand diagram below. \includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-2_698_920_1009_239}
3. Use set notation to define \(C _ { 2 }\).
4. Determine whether the complex number \(1.2 + 0.8\) is in \(C _ { 2 }\).

4 Draw the region in an Argand diagram for which \(| z | \leq 2\) and \(| z | > | z - 3 i |\).

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 $$| z | \geqslant | z - 2 \mathrm { i } |$$

1

1. Sketch on an Argand diagram the locus of points satisfying the equation $$| z - 4 + 3 \mathrm { i } | = 5$$
2. 1. Indicate on your diagram the point \(P\) representing \(z _ { 1 }\), where both $$\left| z _ { 1 } - 4 + 3 \mathrm { i } \right| = 5 \quad \text { and } \quad \arg z _ { 1 } = 0$$
   2. Find the value of \(\left| z _ { 1 } \right|\).

2

1. Draw on an Argand diagram the locus \(L\) of points satisfying the equation \(\arg z = \frac { \pi } { 6 }\).
   (1 mark)
2. 1. A circle \(C\), of radius 6, has its centre lying on \(L\) and touches the line \(\operatorname { Re } ( z ) = 0\). Draw \(C\) on your Argand diagram from part (a).
   2. Find the equation of \(C\), giving your answer in the form \(\left| z - z _ { 0 } \right| = k\).
   3. The complex number \(z _ { 1 }\) lies on \(C\) and is such that \(\arg z _ { 1 }\) has its least possible value. Find \(\arg z _ { 1 }\), giving your answer in the form \(p \pi\), where \(- 1 < p \leqslant 1\).

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.