4.02k Argand diagrams: geometric interpretation

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

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

13 The complex number \(z\) is defined as \(z = \frac { 1 } { 3 } \mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 < \theta < \frac { 1 } { 2 } \pi\).
On an Argand diagram, the point O represents the complex number 0 , and the points \(P _ { 1 } , P _ { 2 } , P _ { 3 } , \ldots\) represent the complex numbers \(z , z ^ { 2 } , z ^ { 3 } , \ldots\) respectively.

1. Write down each of the following.
   1. The ratio of the lengths \(\mathrm { OP } _ { n + 1 } : \mathrm { OP } _ { n }\)
   2. The angle \(\mathrm { P } _ { n + 1 } \mathrm { OP } _ { n }\)
2. 1. Show that \(\left( 3 - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( 3 - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = \mathrm { a } + \mathrm { b } \cos \theta\), where \(a\) and \(b\) are integers to be determined.
   2. By considering the sum to infinity of the series \(z + z ^ { 2 } + z ^ { 3 } + \ldots\), show that $$\frac { 1 } { 3 } \sin \theta + \frac { 1 } { 9 } \sin 2 \theta + \frac { 1 } { 27 } \sin 3 \theta + \ldots = \frac { 3 \sin \theta } { 10 - 6 \cos \theta } .$$

10

1. Show on an Argand diagram the points representing the three cube roots of unity.
2. 1. Find the exact roots of the equation \(z ^ { 3 } - 1 = \sqrt { 3 } \mathrm { i }\), expressing them in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta < \pi\).
   2. The points representing the cube roots of unity form a triangle \(\Delta _ { 1 }\). The points representing the roots of the equation \(z ^ { 3 } - 1 = \sqrt { 3 } \mathrm { i }\) form a triangle \(\Delta _ { 2 }\). State a sequence of two transformations that maps \(\Delta _ { 1 }\) onto \(\Delta _ { 2 }\).
   3. The three roots in part (b)(i) are \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\). By simplifying \(z _ { 1 } + z _ { 2 } + z _ { 3 }\), verify that the sum of these roots is zero.
   4. Hence show that \(\sin 20 ^ { \circ } + \sin 140 ^ { \circ } = \sin 100 ^ { \circ }\).

12 Fig. 12 shows a rhombus OACB in an Argand diagram. The points A and B represent the complex numbers \(z\) and \(w\) respectively. \begin{figure}[h] \end{figure} Prove that \(\arg ( z + w ) = \frac { 1 } { 2 } ( \arg z + \arg w )\).
[0pt] [A copy of Fig. 12 is provided in the Printed Answer Booklet.]

9. The complex numbers \(z\) and \(w\) are represented by the points \(P ( x , y )\) and \(Q ( u , v )\) respectively in Argand diagrams and $$w = z ^ { 2 } - 1$$

1. Show that \(v = 2 x y\) and obtain an expression for \(u\) in terms of \(x\) and \(y\).
2. The point \(P\) moves along the line \(y = 3 x\). Find the equation of the locus of \(Q\).

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.

9. The complex numbers \(z\) and \(w\) are represented by the points \(P ( x , y )\) and \(Q ( u , v )\) respectively, in Argand diagrams, and \(w = 1 - z ^ { 2 }\).

1. Find expressions for \(u\) and \(v\) in terms of \(x\) and \(y\).
2. The point \(P\) moves along the line \(y = 4 x\). Find the equation of the locus of \(Q\).
3. Find the perpendicular distance of the point corresponding to \(z = 2 + 5 \mathrm { i }\) in the \(( u , v )\)-plane, from the locus of \(Q\).

1. The complex numbers \(z\) and \(w\) are represented, respectively, by the points \(P ( x , y )\) and \(Q ( u , v )\) in Argand diagrams and

$$w = \frac { z } { 1 - z }$$

1. Show that \(v = \frac { y } { ( 1 - x ) ^ { 2 } + y ^ { 2 } }\) and obtain an expression for \(u\) in terms of \(x\) and \(y\).
   
2. The point \(P\) moves along the line \(y = 1 - x\). Find and simplify the equation of the locus of \(Q\).
   

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.
   

7. $$f ( z ) = z ^ { 3 } + z ^ { 2 } + p z + q$$ where \(p\) and \(q\) are real constants.
The equation \(f ( z ) = 0\) has roots \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) When plotted on an Argand diagram, the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) form the vertices of a triangle of area 35 Given that \(z _ { 1 } = 3\), find the values of \(p\) and \(q\).

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}

7. $$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(a\), \(b\), \(c\) and \(d\) are real constants.
The equation \(\mathrm { f } ( \mathrm { z } ) = 0\) has complex roots \(\mathrm { z } _ { 1 } , \mathrm { z } _ { 2 } , \mathrm { z } _ { 3 }\) and \(\mathrm { z } _ { 4 }\) When plotted on an Argand diagram, the points representing \(z _ { 1 } , z _ { 2 } , z _ { 3 }\) and \(z _ { 4 }\) form the vertices of a square, with one vertex in each quadrant.
Given that \(z _ { 1 } = 2 + 3 i\), determine the values of \(a , b , c\) and \(d\).

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. \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + a \mathrm { z } ^ { 2 } + b \mathrm { z } + 175 \quad\) where \(a\) and \(b\) are real constants

Given that \(- 3 + 4 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. determine the value of \(a\) and the value of \(b\).
2. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.
3. Write down the roots of the equation \(\mathrm { f } ( \mathrm { z } + 2 ) = 0\)

Given that on an Argand diagram the locus of points defined by \(| z + 5 - 12 i | = 10\) is a circle,

1. write down,
   1. the coordinates of the centre of this circle,
   2. the radius of this circle.
2. Show, by shading on an Argand diagram, the set of points defined by $$| z + 5 - 12 i | \leqslant 10$$
3. For the set of points defined in part (b), determine the maximum value of \(| z |\) The set of points \(A\) is defined by $$A = \{ z : 0 \leqslant \arg ( z + 5 - 20 i ) \leqslant \pi \} \cap \{ z : | z + 5 - 12 i | \leqslant 10 \}$$
4. Determine the area of the region defined by \(A\), giving your answer to 3 significant figures.

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 $$| z - 1 - 8 i | \geqslant 3 | z - 1 | \quad \text { and } \quad 0 \leqslant \arg ( z + i ) \leqslant \frac { \pi } { 4 }$$

5. \begin{figure}[h] \end{figure} Figure 1 shows an Argand diagram.
The set of points, \(A\), that lies within the shaded region, including its boundaries, is defined by $$A = \{ z : p \leqslant \arg ( z ) \leqslant q \} \cap \{ z : | z | \leqslant r \}$$ where \(p , q\) and \(r\) are positive constants.

1. Write down the values of \(p , q\) and \(r\). Given that \(w = - 2 \sqrt { 3 } + 2 \mathrm { i }\) and \(\mathrm { z } \in A\),
2. find the maximum value of \(| w - z | ^ { 2 }\) giving your answer in an exact simplified form.

1. Sketch on an Argand diagram the region defined by

$$z \in \mathbb { C } : - \frac { \pi } { 4 } < \arg ( z + 2 ) < \frac { \pi } { 4 } \cap \{ z \in \mathbb { C } : - 1 < \operatorname { Re } ( z ) \leqslant 1 \}$$ On your sketch

• shade the part of the diagram that is included in the region
• use solid lines to show the parts of the boundary that are included in the region
• use dashed lines to show the parts of the boundary that are not included in the region

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

Given that $$\arg \left( \frac { z - 4 - i } { z - 2 - 7 i } \right) = \frac { \pi } { 2 }$$

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