4.02k Argand diagrams: geometric interpretation

1. A circle \(C\) in the complex plane has equation

$$| z - ( - 3 + 3 i ) | = \alpha | z - ( 1 + 3 i ) |$$ where \(\alpha\) is a real constant with \(\alpha > 1\) Given that the imaginary axis is a tangent to \(C\)

1. sketch, on an Argand diagram, the circle \(C\)
2. explain why the value of \(\alpha\) is 3 The circle \(C\) is contained in the region $$R = \left\{ z \in \mathbb { C } : \beta \leqslant \arg z \leqslant \frac { \pi } { 2 } \right\}$$
3. Determine the maximum value of \(\beta\) Give your answer in radians to 3 significant figures.

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

$$| z - 9 + 12 i | = 2 | z |$$

1. Show that \(C\) is a circle, and find its centre and radius.
2. Sketch \(C\) on an Argand diagram. Given that \(w\) lies on \(C\),
3. find the largest value of \(a\) and the smallest value of \(b\) that must satisfy $$a \leqslant \operatorname { Re } ( w ) \leqslant b$$

1. $$f ( z ) = 3 z ^ { 3 } + p z ^ { 2 } + 57 z + q$$ where \(p\) and \(q\) are real constants.
Given that \(3 - 2 \sqrt { 2 } \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. show all the roots of \(f ( z ) = 0\) on a single Argand diagram,
2. find the value of \(p\) and the value of \(q\).

1. $$\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + a \mathrm { z } + 52 \quad \text { where } a \text { is a real constant }$$ Given that \(2 - 3 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. write down the other complex root.
2. Hence
   1. solve completely \(\mathrm { f } ( \mathrm { z } ) = 0\)
   2. determine the value of \(a\)
3. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

$$z _ { 1 } = - 4 + 4 i$$

1. Express \(\mathrm { z } _ { 1 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r \in \mathbb { R } , r > 0\) and \(0 \leqslant \theta < 2 \pi\) $$z _ { 2 } = 3 \left( \cos \frac { 17 \pi } { 12 } + i \sin \frac { 17 \pi } { 12 } \right)$$
2. Determine in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact real numbers,
   1. \(\frac { Z _ { 1 } } { Z _ { 2 } }\)
   2. \(\left( z _ { 2 } \right) ^ { 4 }\)
3. Show on a single Argand diagram
   1. the complex numbers \(z _ { 1 } , z _ { 2 }\) and \(\frac { z _ { 1 } } { z _ { 2 } }\)
   2. the region defined by \(\left\{ z \in \mathbb { C } : \left| z - z _ { 1 } \right| < \left| z - z _ { 2 } \right| \right\}\)

1. $$\mathrm { f } ( z ) = z ^ { 4 } - 6 z ^ { 3 } + a z ^ { 2 } + b z + 145$$ where \(a\) and \(b\) are real constants.
Given that \(2 + 5 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. determine the other roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
2. Show all the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

3. $$\mathrm { f } ( z ) = z ^ { 4 } + a z ^ { 3 } + 6 z ^ { 2 } + b z + 65$$ where \(a\) and \(b\) are real constants.
Given that \(z = 3 + 2 \mathbf { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\), show the roots of \(\mathrm { f } ( z ) = 0\) on a single Argand diagram.

In an Argand diagram, the points \(A , B\) and \(C\) are the vertices of an equilateral triangle with its centre at the origin. The point \(A\) represents the complex number \(6 + 2 \mathrm { i }\).

1. Find the complex numbers represented by the points \(B\) and \(C\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact. The points \(D , E\) and \(F\) are the midpoints of the sides of triangle \(A B C\).
2. Find the exact area of triangle \(D E F\).

In an Argand diagram, the points \(A\) and \(B\) are represented by the complex numbers \(- 3 + 2 \mathrm { i }\) and \(5 - 4 \mathrm { i }\) respectively. The points \(A\) and \(B\) are the end points of a diameter of a circle \(C\).

1. Find the equation of \(C\), giving your answer in the form $$| z - a | = b \quad a \in \mathbb { C } , \quad b \in \mathbb { R }$$ The circle \(D\), with equation \(| z - 2 - 3 i | = 2\), intersects \(C\) at the points representing the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\)
2. Find the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\)

1. The point \(P\) is one vertex of a regular pentagon in an Argand diagram. The centre of the pentagon is at the origin.
   Given that \(P\) represents the complex number \(6 + 6 \mathrm { i }\), determine the complex numbers that represent the other vertices of the pentagon, giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\)
2. (a) On a single Argand diagram, shade the region, \(R\), that satisfies both $$| z - 2 i | \leqslant 2 \quad \text { and } \quad \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 3 } \pi$$

(b) Determine the exact area of \(R\), giving your answer in simplest form.

1. The points representing the complex numbers \(z _ { 1 } = 35 - 25 i\) and \(z _ { 2 } = - 29 + 39 i\) are opposite vertices of a regular hexagon, \(H\), in the complex plane.

The centre of \(H\) represents the complex number \(\alpha\)

1. Show that \(\alpha = 3 + 7 \mathrm { i }\) Given that \(\beta = \frac { 1 + \mathrm { i } } { 64 }\)
2. show that $$\beta \left( z _ { 1 } - \alpha \right) = 1$$ The vertices of \(H\) are given by the roots of the equation $$( \beta ( z - \alpha ) ) ^ { 6 } = 1$$
3. 1. Write down the roots of the equation \(w ^ { 6 } = 1\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\)
   2. Hence, or otherwise, determine the position of the other four vertices of \(H\), giving your answers as complex numbers in Cartesian form.

1. The locus \(C\) is given by

$$| z - 4 | = 4$$ The locus \(D\) is given by $$\arg z = \frac { \pi } { 3 }$$

1. Sketch, on the same Argand diagram, the locus \(C\) and the locus \(D\) The set of points \(A\) is defined by $$A = \{ z \in \mathbb { C } : | z - 4 | \leqslant 4 \} \cap \left\{ z \in \mathbb { C } : 0 \leqslant \arg z \leqslant \frac { \pi } { 3 } \right\}$$
2. Show, by shading on your Argand diagram, the set of points \(A\)
3. Find the area of the region defined by \(A\), giving your answer in the form \(p \pi + q \sqrt { 3 }\) where \(p\) and \(q\) are constants to be determined.

1. A complex number \(z = x + \mathrm { i } y\) is represented by the point \(P\) in an Argand diagram.

Given that $$| z - 3 | = 4 | z + 1 |$$

1. show that the locus of \(P\) has equation $$15 x ^ { 2 } + 15 y ^ { 2 } + 38 x + 7 = 0$$
2. Hence find the maximum value of \(| z |\)

1. A transformation from the \(z\)-plane to the \(w\)-plane is given by

$$w = \frac { 3 \mathrm { i } z - 2 } { z + \mathrm { i } } \quad z \neq - \mathrm { i }$$

1. Show that the circle \(C\) with equation \(| z + \mathrm { i } | = 1\) in the \(z\)-plane is mapped to a circle \(D\) in the \(w\)-plane, giving a Cartesian equation for \(D\).
2. Sketch \(C\) and \(D\) on Argand diagrams.
   

1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by

$$w = \frac { 1 - 3 z } { z + 2 i } \quad z \neq - 2 i$$ The circle with equation \(| z + \mathrm { i } | = 3\) is mapped by \(T\) onto the circle \(C\).

1. Show that the equation for \(C\) can be written as $$3 | w + 3 | = | 1 + ( 3 - w ) \mathrm { i } |$$
2. Hence find
   1. a Cartesian equation for \(C\),
   2. the centre and radius of \(C\).

1. The point \(P\) in the complex plane represents a complex number \(z\) such that

$$| z + 9 | = 4 | z - 12 i |$$ Given that, as \(z\) varies, the locus of \(P\) is a circle,

1. determine the centre and radius of this circle.
2. Shade on an Argand diagram the region defined by the set $$\{ z \in \mathbb { C } : | z + 9 | < 4 | z - 12 i | \} \cap \left\{ z \in \mathbb { C } : - \frac { \pi } { 4 } < \arg \left( z - \frac { 3 + 44 i } { 5 } \right) < \frac { \pi } { 4 } \right\}$$

1. The locus of points \(z\) satisfies

$$| z + a \mathrm { i } | = 3 | z - a |$$ where \(a\) is an integer.
The locus is a circle with its centre in the third quadrant and radius \(\frac { 3 } { 2 } \sqrt { 2 }\) Determine

1. the value of \(a\),
2. the coordinates of the centre of the circle.

1. The locus of points \(z = x + \mathrm { i } y\) that satisfy

$$\arg \left( \frac { z - 8 - 5 i } { z - 2 - 5 i } \right) = \frac { \pi } { 3 }$$ is an arc of a circle \(C\).

1. On an Argand diagram sketch the locus of \(z\).
2. Explain why the centre of \(C\) has \(x\) coordinate 5
3. Determine the radius of \(C\).
4. Determine the \(y\) coordinate of the centre of \(C\).

1. A complex number \(z\) is represented by the point \(P\) in the complex plane.

Given that \(z\) satisfies $$| z - 6 | = 2 | z + 3 i |$$

1. show that the locus of \(P\) passes through the origin and the points - 4 and - 8 i
2. Sketch on an Argand diagram the locus of \(P\) as \(z\) varies.
3. On your sketch, shade the region which satisfies both $$| z - 6 | \geqslant 2 | z + 3 i | \text { and } | z | \leqslant 4$$

9. \begin{figure}[h] \end{figure} Figure 1 shows a locus in the complex plane.
The locus is an arc of a circle from the point represented by \(z _ { 1 } = 3 + 2 i\) to the point represented by \(z _ { 2 } = a + 4 \mathrm { i }\), where \(a\) is a constant, \(a \neq 1\) Given that

• the point \(z _ { 3 } = 1 + 4 \mathrm { i }\) also lies on the locus
• the centre of the circle has real part equal to - 1
  1. determine the value of \(a\).
  2. Hence determine a complex equation for the locus, giving any angles in the equation as positive values.

1. A curve has equation

$$| z + 6 | = 2 | z - 6 | \quad z \in \mathbb { C }$$

1. Show that the curve is a circle with equation \(x ^ { 2 } + y ^ { 2 } - 20 x + 36 = 0\)
2. Sketch the curve on an Argand diagram. The line \(l\) has equation \(a z ^ { * } + a ^ { * } z = 0\), where \(a \in \mathbb { C }\) and \(z \in \mathbb { C }\) Given that the line \(l\) is a tangent to the curve and that \(\arg a = \theta\)
3. find the possible values of \(\tan \theta\)