4.02k Argand diagrams: geometric interpretation

4. $$z _ { 1 } = 3 \mathrm { i } \text { and } z _ { 2 } = \frac { 6 } { 1 + \mathrm { i } \sqrt { 3 } }$$

1. Express \(z _ { 2 }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers.
2. Find the modulus and the argument of \(z _ { 2 }\), giving the argument in radians in terms of \(\pi\).
3. Show the three points representing \(z _ { 1 } , z _ { 2 }\) and \(\left( z _ { 1 } + z _ { 2 } \right)\) respectively, on a single Argand diagram.

7. A complex number \(z\) is given by $$z = a + 2 i$$ where \(a\) is a non-zero real number.

1. Find \(z ^ { 2 } + 2 z\) in the form \(x +\) iy where \(x\) and \(y\) are real expressions in terms of \(a\). Given that \(z ^ { 2 } + 2 z\) is real,
2. find the value of \(a\). Using this value for \(a\),
3. find the values of the modulus and argument of \(z\), giving the argument in radians, and giving your answers to 3 significant figures.
4. Show the points \(P , Q\) and \(R\), representing the complex numbers \(z , z ^ { 2 }\) and \(z ^ { 2 } + 2 z\) respectively, on a single Argand diagram with origin \(O\).
5. Describe fully the geometrical relationship between the line segments \(O P\) and \(Q R\).
   

7. The quadratic equation $$z ^ { 2 } + 10 z + 169 = 0$$ has complex roots \(z _ { 1 }\) and \(z _ { 2 }\).

1. Find each of these roots in the form \(a + b \mathrm { i }\).
2. Find the modulus and argument of \(z _ { 1 }\) and of \(z _ { 2 }\). Give the arguments in radians to 3 significant figures.
3. Illustrate the two roots on a single Argand diagram.
4. Find the value of \(\left| z _ { 1 } - z _ { 2 } \right|\).

6. Given that \(z = - 3 + 4 \mathrm { i }\),

1. find the modulus of \(z\),
2. the argument of \(z\) in radians to 2 decimal places. Given also that \(w = \frac { - 14 + 2 \mathrm { i } } { z }\),
3. use algebra to find \(w\), giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real. The complex numbers \(z\) and \(w\) are represented by points \(A\) and \(B\) on an Argand diagram.
4. Show the points \(A\) and \(B\) on an Argand diagram.

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

Given that $$| z - 2 i | = | z - 3 |$$

1. sketch the locus of \(P\). You do not need to find the coordinates of any intercepts. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { \mathrm { i } z } { z - 2 \mathrm { i } } \quad z \neq 2 \mathrm { i }$$ Given that \(T\) maps \(| z - 2 i | = | z - 3 |\) to a circle \(C\) in the \(w\)-plane,
2. find the equation of \(C\), giving your answer in the form $$| w - ( p + q \mathrm { i } ) | = r$$ where \(p , q\) and \(r\) are real numbers to be determined.

1. A 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

$$w = \frac { z - 3 } { 2 \mathrm { i } - z } \quad z \neq 2 \mathrm { i }$$ The line in the \(z\)-plane with equation \(y = x + 3\) is mapped by \(T\) onto a circle \(C\) in the \(w\)-plane.

1. Determine
   1. the coordinates of the centre of \(C\)
   2. the exact radius of \(C\) The region \(y > x + 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
2. On a single Argand diagram
   1. sketch the circle \(C\)
   2. shade and label the region \(R\)

4. A transformation from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z } { z + 3 } , \quad z \neq - 3$$ Under this transformation, the circle \(| z | = 2\) in the \(z\)-plane is mapped onto a circle \(C\) in the \(w\)-plane. Determine the centre and the radius of the circle \(C\).

7. The point \(P\) represents a complex number \(z\) on an Argand diagram, where $$| z + 1 | = | 2 z - 1 |$$ and the point \(Q\) represents a complex number \(w\) on the Argand diagram, where $$| w | = | w - 1 + \mathrm { i } |$$ Find the exact coordinates of the points where the locus of \(P\) intersects the locus of \(Q\).

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 \(R\).

5. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 2 z - 1 } { z + 3 } , \quad z \neq - 3$$ The circle in the \(z\)-plane with equation \(x ^ { 2 } + y ^ { 2 } = 1\), where \(z = x + \mathrm { i } y\), is mapped by \(T\) onto the circle \(C\) in the \(w\)-plane. Find the centre and the radius of \(C\).

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

1. the centre of \(C\),
2. the radius of \(C\).
   

2. 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 $$w = \frac { z + 2 } { z - \mathrm { i } } \quad z \neq \mathrm { i }$$ The transformation \(T\) maps the circle \(| z | = 2\) in the \(z\)-plane onto a circle \(C\) in the \(w\)-plane.
Find (i) the centre of \(C\),
(ii) the radius of \(C\).

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

$$w = \frac { z } { z + 4 \mathrm { i } } \quad z \neq - 4 \mathrm { i }$$ The circle with equation \(| z | = 3\) is mapped by \(T\) onto the circle \(C\) Determine

1. a Cartesian equation of \(C\)
2. the centre and radius of \(C\)

1. The complex number \(z _ { 1 }\) is defined as

$$z _ { 1 } = \frac { \left( \cos \frac { 5 \pi } { 12 } + i \sin \frac { 5 \pi } { 12 } \right) ^ { 4 } } { \left( \cos \frac { \pi } { 3 } - i \sin \frac { \pi } { 3 } \right) ^ { 3 } }$$

1. Without using your calculator show that $$z _ { 1 } = \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 }$$
2. Shade, on a single Argand diagram, the region \(R\) defined by $$\left| z - z _ { 1 } \right| \leqslant 1 \quad \text { and } \quad 0 \leqslant \arg \left( z - z _ { 1 } \right) \leqslant \frac { 3 \pi } { 4 }$$ Given that the complex number \(z\) lies in \(R\)
3. determine the smallest possible positive value of \(\arg z\)

1. 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

$$w = \frac { z + 1 } { z - 3 } \quad z \neq 3$$ The straight line in the \(z\)-plane with equation \(y = 4 x\) is mapped by \(T\) onto the circle \(C\) in the \(w\)-plane.

1. Show that \(C\) has equation $$3 u ^ { 2 } + 3 v ^ { 2 } - 2 u + v + k = 0$$ where \(k\) is a constant to be determined.
2. Hence determine
   1. the coordinates of the centre of \(C\)
   2. the radius of \(C\)

1. The complex number \(z = x + i y\) satisfies the equation

$$| z - 3 - 4 i | = | z + 1 + i |$$

1. Determine an equation for the locus of \(z\) giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
2. Shade, on an Argand diagram, the region defined by $$| z - 3 - 4 i | \leqslant | z + 1 + i |$$ You do not need to determine the coordinates of any intercepts on the coordinate axes.

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

$$w = \frac { z - \mathrm { i } } { z + 1 } \quad z \neq - 1$$ Given that \(T\) maps the imaginary axis in the \(z\)-plane to the circle \(C\) in the \(w\)-plane, determine (i) the coordinates of the centre of \(C\) (ii) the radius of \(C\)

8. In the Argand diagram the point \(P\) represents the complex number \(z\). Given that arg \(\left( \frac { z - 2 \mathrm { i } } { z + 2 } \right) = \frac { \pi } { 2 }\),

1. sketch the locus of \(P\),
2. deduce the value of \(| \mathrm { z } + 1 - \mathrm { i } |\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is defined by $$w = \frac { 2 ( 1 + \mathrm { i } ) } { z + 2 } , \quad z \neq - 2$$
3. Show that the locus of \(P\) in the \(z\)-plane is mapped to part of a straight line in the \(w\)-plane, and show this in an Argand diagram.
   (6)(Total 12 marks)

9. (a) The point \(P\) represents a complex number \(z\) in an Argand diagram. Given that $$| z - 2 i | = 2 | z + i |$$

1. find a cartesian equation for the locus of \(P\), simplifying your answer.
2. sketch the locus of \(P\).
   (b) A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is a translation \(- 7 + 11\) i followed by an enlargement with centre the origin and scale factor 3 . Write down the transformation \(T\) in the form $$w = a z + b , \quad a , b \in \mathbb { C }$$

1. (i) (a) On the same Argand diagram sketch the loci given by the following equations.

$$| z - 1 | = 1 , \quad , , \arg ( z + 1 ) = \frac { \pi } { 12 } , \quad , \arg ( z + 1 ) = \frac { \pi } { 2 }$$ (b) Shade on your diagram the region for which $$| z - 1 | \leq 1 \quad \text { and } \quad \frac { \pi } { 12 } \leq \arg ( z + 1 ) \leq \frac { \pi } { 2 }$$ (ii) (a) Show that the transformation \(\quad w = \frac { z - 1 } { z } , \quad z \neq 0\), $$\text { maps } | z - 1 | = 1 \text { in the } \boldsymbol { z } \text {-plane onto } | w | = | w - 1 | \text { in the } \boldsymbol { w } \text {-plane. }$$ The region \(| z - 1 | \leq 1\) in the \(z\)-plane is mapped onto the region \(T\) in the \(w\)-plane.
(b) Shade the region \(T\) on an Argand diagram.

12. The transformation \(T\) from the complex \(z\)-plane to the complex \(w\)-plane is given by $$w = \frac { z + 1 } { z + \mathrm { i } } , \quad z \neq - \mathrm { i }$$

1. Show that \(T\) maps points on the half-line \(\arg ( z ) = \frac { \pi } { 4 }\) in the \(z\)-plane into points on the circle \(| w | = 1\) in the \(w\)-plane.
2. Find the image under \(T\) in the \(w\)-plane of the circle \(| Z | = 1\) in the \(z\)-plane.
3. Sketch on separate diagrams the circle \(| \mathbf { Z } | = 1\) in the \(z\)-plane and its image under \(T\) in the \(w\)-plane.
4. Mark on your sketches the point \(P\), where \(z = \mathrm { i }\), and its image \(Q\) under \(T\) in the \(w\)-plane.

9. A complex number \(z\) is represented by the point \(P\) in the Argand diagram. Given that $$| z - 3 \mathrm { i } | = 3$$

1. sketch the locus of \(P\).
2. Find the complex number \(z\) which satisfies both \(| z - 3 i | = 3\) and \(\arg ( z - 3 i ) = \frac { 3 } { 4 } \pi\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 2 \mathrm { i } } { z }$$
3. Show that \(T\) maps \(| z - 3 i | = 3\) to a line in the \(w\)-plane, and give the cartesian equation of this line.
   (5)(Total 11 marks)

8. The point \(P\) represents a complex number \(z\) on an Argand diagram, where $$| z - 6 + 3 i | = 3 | z + 2 - i |$$

1. Show that the locus of \(P\) is a circle, giving the coordinates of the centre and the radius of this circle. The point \(Q\) represents a complex number \(z\) on an Argand diagram, where $$\tan [ \arg ( z + 6 ) ] = \frac { 1 } { 2 }$$
2. On the same Argand diagram, sketch the locus of \(P\) and the locus of \(Q\).
3. On your diagram, shade the region which satisfies both $$| z - 6 + 3 \mathrm { i } | > 3 | z + 2 - \mathrm { i } | \text { and } \tan [ \arg ( z + 6 ) ] > \frac { 1 } { 2 }$$ (2)(Total 14 marks)

1. 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

$$w = \frac { z + \mathrm { i } } { \mathrm { z } } , \quad z \neq 0 .$$

1. The transformation \(T\) maps the points on the line with equation \(y = x\) in the \(z\)-plane, other than \(( 0,0 )\), to points on a line \(l\) in the \(w\)-plane. Find a cartesian equation of \(l\).
2. Show that the image, under \(T\), of the line with equation \(x + y + 1 = 0\) in the \(z\)-plane is a circle \(C\) in the \(w\)-plane, where \(C\) has cartesian equation $$u ^ { 2 } + v ^ { 2 } - u + v = 0$$
3. On the same Argand diagram, sketch \(l\) and \(C\).

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

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

1. Show that \(C\) is a circle and find its centre and radius. The region \(| z | < 3\) 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.