4.02k Argand diagrams: geometric interpretation

6. A complex number \(z\) is represented by the point \(P\) in the Argand diagram.

1. Given that \(| z - 6 | = | z |\), sketch the locus of \(P\).
2. Find the complex numbers \(z\) which satisfy both \(| z - 6 | = | z |\) and \(| z - 3 - 4 \mathrm { i } | = 5\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \(w = \frac { 30 } { z }\).
3. Show that \(T\) maps \(| z - 6 | = | z |\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle.

1. The point \(P\) represents the complex number \(z\) on an Argand diagram, where

$$| z - \mathrm { i } | = 2$$ The locus of \(P\) as \(z\) varies is the curve \(C\).

1. Find a cartesian equation of \(C\).
2. Sketch the curve \(C\). A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z + \mathrm { i } } { 3 + \mathrm { i } z } , \quad z \neq 3 \mathrm { i }$$ The point \(Q\) is mapped by \(T\) onto the point \(R\). Given that \(R\) lies on the real axis,
3. show that \(Q\) lies on \(C\).
   

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 \(\arg ( z - 6 ) = - \frac { 3 \pi } { 4 }\)

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

$$w = \frac { z + 2 \mathrm { i } } { \mathrm { i } z } \quad z \neq 0$$ The transformation maps points on the real axis in the \(z\)-plane onto a line in the \(w\)-plane. Find an equation of this line.

6. Solve the equation $$z ^ { 5 } = - 16 \sqrt { } 3 + 16 i$$ giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta < \pi\).

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

1. Given that \(| z | = 1\), sketch the locus of \(P\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z + 7 \mathrm { i } } { z - 2 \mathrm { i } }$$
2. Show that \(T\) maps \(| z | = 1\) onto a circle in the \(w\)-plane.
3. Show that this circle has its centre at \(w = - 5\) and find its radius.

6. The transformation \(T\) maps points from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\). The transformation \(T\) is given by $$w = \frac { z } { i z + 1 } , \quad z \neq i$$ The transformation \(T\) maps the line \(l\) in the \(z\)-plane onto the line with equation \(v = - 1\) in the \(w\)-plane.

1. Find a cartesian equation of \(l\) in terms of \(x\) and \(y\). The transformation \(T\) maps the line with equation \(y = \frac { 1 } { 2 }\) in the \(z\)-plane onto the curve \(C\) in the \(w\)-plane.
2. 1. Show that \(C\) is a circle with centre the origin.
   2. Write down a cartesian equation of \(C\) in terms of \(u\) and \(v\).

6. The transformation \(T\) from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\), is given by $$w = \frac { 4 ( 1 - \mathrm { i } ) z - 8 \mathrm { i } } { 2 ( - 1 + \mathrm { i } ) z - \mathrm { i } } , \quad z \neq \frac { 1 } { 4 } - \frac { 1 } { 4 } \mathrm { i }$$ The transformation \(T\) maps the points on the line \(l\) with equation \(y = x\) in the \(z\)-plane to a circle \(C\) in the \(w\)-plane.

1. Show that $$w = \frac { a x ^ { 2 } + b x i + c } { 16 x ^ { 2 } + 1 }$$ where \(a\), \(b\) and \(c\) are real constants to be found.
2. Hence show that the circle \(C\) has equation $$( u - 3 ) ^ { 2 } + v ^ { 2 } = k ^ { 2 }$$ where \(k\) is a constant to be found.

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

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

1. 1. Show that \(C\) is a circle.
   2. Find the centre and radius of \(C\). The region \(| z | \leqslant 2\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
2. Shade the region \(R\) on an Argand diagram.

3.

1. Find the four roots of the equation \(z ^ { 4 } = 8 ( \sqrt { 3 } + \mathrm { i } )\) in the form \(z = r \mathrm { e } ^ { \mathrm { i } \theta }\)
2. Show these roots on an Argand diagram.
   

8. A complex number \(z\) satisfies the equation $$| z - 5 - 12 i | = 3$$

1. Describe in geometrical terms with the aid of a sketch, the locus of the point which represents \(z\) in the A rgand diagram. For points on this locus, find
2. the maximum and minimum values for \(| z |\),
3. the maximum and minimum values for arg \(z\), giving your answers in radians to 2 decimal places.

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 \(R\).

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

$$w = \frac { z - 1 } { z + 1 } , \quad z \neq - 1$$ The line in the \(z\)-plane with equation \(y = 2 x\) is mapped by \(T\) onto the curve \(C\) in the \(w\)-plane.

1. Show that \(C\) is a circle and find its centre and radius. The region \(y < 2 x\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
2. Sketch circle \(C\) on an Argand diagram and shade and label region \(R\).

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

1. Given that \(| z - 6 | = | z |\), sketch the locus of \(P\).
2. Find the complex numbers \(z\) which satisfy both \(| z - 6 | = | z |\) and \(| z - 3 - 4 \mathrm { i } | = 5\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \(w = \frac { 30 } { z }\).
3. Show that \(T\) maps \(| z - 6 | = | z |\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle.

2

1. 1. Given that \(z = \cos \theta + \mathrm { j } \sin \theta\), express \(z ^ { n } + \frac { 1 } { z ^ { n } }\) and \(z ^ { n } - \frac { 1 } { z ^ { n } }\) in simplified trigonometric form.
   2. By considering \(\left( z - \frac { 1 } { z } \right) ^ { 4 } \left( z + \frac { 1 } { z } \right) ^ { 2 }\), find \(A , B , C\) and \(D\) such that $$\sin ^ { 4 } \theta \cos ^ { 2 } \theta = A \cos 6 \theta + B \cos 4 \theta + C \cos 2 \theta + D$$
2. 1. Find the modulus and argument of \(4 + 4 \mathrm { j }\).
   2. Find the fifth roots of \(4 + 4 \mathrm { j }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Illustrate these fifth roots on an Argand diagram.
   3. Find integers \(p\) and \(q\) such that \(( p + q \mathrm { j } ) ^ { 5 } = 4 + 4 \mathrm { j }\).

2

1. Use de Moivre's theorem to show that \(\sin 5 \theta = 5 \sin \theta - 20 \sin ^ { 3 } \theta + 16 \sin ^ { 5 } \theta\).
2. 1. Find the cube roots of \(- 2 + 2 \mathrm { j }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). These cube roots are represented by points \(\mathrm { A } , \mathrm { B }\) and C in the Argand diagram, with A in the first quadrant and ABC going anticlockwise. The midpoint of AB is M , and M represents the complex number \(w\).
   2. Draw an Argand diagram, showing the points \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and M .
   3. Find the modulus and argument of \(w\).
   4. Find \(w ^ { 6 }\) in the form \(a + b \mathrm { j }\).

2

1. Given that \(z = \cos \theta + \mathrm { j } \sin \theta\), express \(z ^ { n } + \frac { 1 } { z ^ { n } }\) and \(z ^ { n } - \frac { 1 } { z ^ { n } }\) in simplified trigonometric form.
   Hence find the constants \(A , B , C\) in the identity $$\sin ^ { 5 } \theta \equiv A \sin \theta + B \sin 3 \theta + C \sin 5 \theta$$
2. 1. Find the 4th roots of - 9 j in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\) and \(0 < \theta < 2 \pi\). Illustrate the roots on an Argand diagram.
   2. Let the points representing these roots, taken in order of increasing \(\theta\), be \(\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S }\). The mid-points of the sides of PQRS represent the 4th roots of a complex number \(w\). Find the modulus and argument of \(w\). Mark the point representing \(w\) on your Argand diagram.

1

1. By writing \(z\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), show that \(z z ^ { * } = | z | ^ { 2 }\).
2. Given that \(z z ^ { * } = 9\), describe the locus of \(z\).

4 In this question, give your answers exactly in polar form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).

1. Express \(4 ( ( \sqrt { } 3 ) - \mathrm { i } )\) in polar form.
2. Find the cube roots of \(4 ( ( \sqrt { } 3 ) - \mathrm { i } )\) in polar form.
3. Sketch an Argand diagram showing the positions of the cube roots found in part (ii). Hence, or otherwise, prove that the sum of these cube roots is zero.

7

1. The complex number \(3 + 2 \mathrm { i }\) is denoted by \(w\) and the complex conjugate of \(w\) is denoted by \(w ^ { * }\). Find
   1. the modulus of \(w\),
   2. the argument of \(w ^ { * }\), giving your answer in radians, correct to 2 decimal places.
2. Find the complex number \(u\) given that \(u + 2 u ^ { * } = 3 + 2 \mathrm { i }\).
3. Sketch, on an Argand diagram, the locus given by \(| z + 1 | = | z |\).

4

1. Sketch, on an Argand diagram, the locus given by \(| z - 1 + \mathrm { i } | = \sqrt { 2 }\).
2. Shade on your diagram the region given by \(1 \leqslant | z - 1 + \mathrm { i } | \leqslant \sqrt { 2 }\).

5

1. Verify that \(z ^ { 3 } - 8 = ( z - 2 ) \left( z ^ { 2 } + 2 z + 4 \right)\).
2. Solve the quadratic equation \(z ^ { 2 } + 2 z + 4 = 0\), giving your answers exactly in the form \(x + \mathrm { i } y\). Show clearly how you obtain your answers.
3. Show on an Argand diagram the roots of the cubic equation \(z ^ { 3 } - 8 = 0\).

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 \(C _ { 2 }\).