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

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}

1

1. Sketch on an Argand diagram the locus of points satisfying the equation $$| z - 6 \mathrm { i } | = 3$$
2. It is given that \(z\) satisfies the equation \(| z - 6 \mathrm { i } | = 3\).
   1. Write down the greatest possible value of \(| z |\).
   2. Find the greatest possible value of \(\arg z\), giving your answer in the form \(p \pi\), where \(- 1 < p \leqslant 1\).

2

1. Sketch, on the Argand diagram below, the locus \(L\) of points satisfying $$\arg ( z - 2 \mathrm { i } ) = \frac { 2 \pi } { 3 }$$
2. 1. A circle \(C\), of radius 3, has its centre lying on \(L\) and touches the line \(\operatorname { Im } ( z ) = 2\). Sketch \(C\) on the Argand diagram used in part (a).
   2. Find the centre of \(C\), giving your answer in the form \(a + b \mathrm { i }\).
      [0pt] [3 marks]

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 \(\arg ( z + 4 ) = \frac { 1 } { 4 } \pi\).

5 An Argand diagram is shown below. The circle has centre at the point representing \(1 + 3 i\), and the half line intersects the circle at the origin and at the point representing \(4 + 4 \mathrm { i }\). \includegraphics[max width=\textwidth, alt={}, center]{c4484913-14bf-4bf4-a290-0301586333ce-3_748_917_351_242} State the two conditions that define the set of complex numbers represented by points in the shaded segment, including its boundaries.

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 \(| z + 2 - 4 i | \geqslant | z - 2 - 6 i |\).

6 On separate Argand diagrams, sketch the set of points represented by each of the following.

1. \(| z - 1 - 2 i | \leqslant 4\).
2. \(\quad \arg ( z + \mathrm { i } ) = \frac { 1 } { 3 } \pi\).

6. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram. Given that $$| z - 1 | = | z - 2 \mathrm { i } |$$ show that the locus of \(P\) is a straight line.

5. (a) The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram. Given that $$| z - 3 + 2 i | = | z - 3 |$$ find the equation of the locus of \(P\).
(b) Give a geometric interpretation of the locus of \(P\).

6. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram. Given that $$| z - 3 + \mathrm { i } | = 2 | z - 5 - 2 \mathrm { i } |$$ show that the locus of \(P\) is a circle and write down the coordinates of its centre.

1. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram.

Two loci, \(L _ { 1 }\) and \(L _ { 2 }\), are given by: $$\begin{aligned} & L _ { 1 } : | z - 2 + \mathrm { i } | = | z + 2 - 3 \mathrm { i } | \\ & L _ { 2 } : | z - 2 + \mathrm { i } | = \sqrt { 10 } \end{aligned}$$

1. Find the coordinates of the points of intersection of these loci.
2. On the same Argand diagram, sketch the loci \(L _ { 1 }\) and \(L _ { 2 }\). Clearly label the coordinates of the points of intersection.

1. (a) Shade on an Argand diagram the set of points

$$\{ z \in \mathbb { C } : | z - 1 - \mathrm { i } | \leqslant 3 \} \cap \quad z \in \mathbb { C } : \frac { \pi } { 4 } \leqslant \arg ( z - 2 ) \leqslant \frac { 3 \pi } { 4 }$$ The complex number \(w\) satisfies $$| w - 1 - \mathrm { i } | = 3 \text { and } \arg ( w - 2 ) = \frac { \pi } { 4 }$$ (b) Find, in simplest form, the exact value of \(| w | ^ { 2 }\)

5. \begin{figure}[h] \end{figure} The complex numbers \(z _ { 1 } = - 2 , z _ { 2 } = - 1 + 2 \mathrm { i }\) and \(z _ { 3 } = 1 + \mathrm { i }\) are plotted in Figure 1, on an Argand diagram for the complex plane with \(z = x + \mathrm { i } y\)

1. Explain why \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) cannot all be roots of a quartic polynomial equation with real coefficients.
2. Show that \(\arg \left( \frac { z _ { 2 } - z _ { 1 } } { z _ { 3 } - z _ { 1 } } \right) = \frac { \pi } { 4 }\)
3. Hence show that \(\arctan ( 2 ) - \arctan \left( \frac { 1 } { 3 } \right) = \frac { \pi } { 4 }\) A copy of Figure 1, labelled Diagram 1, is given on page 12.
4. Shade, on Diagram 1, the set of points of the complex plane that satisfy the inequality $$| z + 2 | \leqslant | z - 1 - \mathrm { i } |$$
   \includegraphics[max width=\textwidth, alt={}]{9312b91c-bca7-4427-a1f7-cb03065ee5e5-12_479_524_296_776}
   \section*{Diagram 1}

1. Given that there are two distinct complex numbers \(z\) that satisfy

$$\{ z : | z - 3 - 5 i | = 2 r \} \cap \quad z : \arg ( z - 2 ) = \frac { 3 \pi } { 4 }$$ determine the exact range of values for the real constant \(r\).

5. \begin{figure}[h] \end{figure} Figure 1 shows an Argand diagram.
The set \(P\), of points that lie within the shaded region including its boundaries, is defined by $$P = \{ z \in \mathbb { C } : a \leqslant | z + b + c \mathrm { i } | \leqslant d \}$$ where \(a\), \(b\), \(c\) and \(d\) are integers.

1. Write down the values of \(a , b , c\) and \(d\). The set \(Q\) is defined by $$Q = \{ z \in \mathbb { C } : a \leqslant | z + b + c \mathrm { i } | \leqslant d \} \cap \{ z \in \mathbb { C } : | z - \mathrm { i } | \leqslant | z - 3 \mathrm { i } | \}$$
2. Determine the exact area of the region defined by \(Q\), giving your answer in simplest form.

1. (i) Shade, on an Argand diagram, the set of points for which

$$| z - 3 | \leqslant | z + 6 i |$$ (ii) Determine the exact complex number \(w\) which satisfies both $$\arg ( w - 2 ) = \frac { \pi } { 3 } \quad \text { and } \quad \arg ( w + 1 ) = \frac { \pi } { 6 }$$

1. (a) Shade on an Argand diagram the set of points

$$\{ z \in \mathbb { C } : | z - 4 i | \leqslant 3 \} \cap \left\{ z \in \mathbb { C } : - \frac { \pi } { 2 } < \arg ( z + 3 - 4 i ) \leqslant \frac { \pi } { 4 } \right\}$$ The complex number \(w\) satisfies $$| w - 4 \mathrm { i } | = 3$$ (b) Find the maximum value of \(\arg w\) in the interval \(( - \pi , \pi ]\). Give your answer in radians correct to 2 decimal places.

1. A complex number \(z\) is represented by the point \(P\) on an Argand diagram.

Given that \(\arg \left( \frac { z - 6 i } { z - 3 i } \right) = \frac { \pi } { 3 }\)

1. sketch the locus of \(P\) as \(z\) varies,
2. find the exact maximum possible value of \(| z |\)

1. A curve \(C\) in the complex plane is described by the equation

$$| z - 1 - 8 i | = 3 | z - 1 |$$

1. Show that \(C\) is a circle, and find its centre and radius.
2. Using the answer to part (a), determine whether \(z = 3 - 3 \mathrm { i }\) satisfies the inequality $$| z - 1 - 8 i | \geqslant 3 | z - 1 |$$
3. Shade, on an Argand diagram, the set of points that satisfies both $$| z - 1 - 8 i | \geqslant 3 | z - 1 | \quad \text { and } \quad 0 \leqslant \arg ( z + i ) \leqslant \frac { \pi } { 4 }$$