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

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 \(| z |\)

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. (i) 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 \(r \mathrm { e } ^ { \mathrm { i } \theta }\) (ii) (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. (i) Given that

$$z _ { 1 } = 6 \mathrm { e } ^ { \frac { \pi } { 3 } \mathrm { i } } \text { and } z _ { 2 } = 6 \sqrt { 3 } \mathrm { e } ^ { \frac { 5 \pi } { 6 } \mathrm { i } }$$ show that $$z _ { 1 } + z _ { 2 } = 12 \mathrm { e } ^ { \frac { 2 \pi } { 3 } \mathrm { i } }$$ (ii) Given that $$\arg ( z - 5 ) = \frac { 2 \pi } { 3 }$$ determine the least value of \(| z |\) as \(z\) varies.

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 \(T\) 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 \(\tan \theta\)

8 Throughout this question the use of a calculator is not permitted.

1. The complex numbers \(u\) and \(v\) satisfy the equations $$u + 2 v = 2 \mathrm { i } \quad \text { and } \quad \mathrm { i } u + v = 3$$ Solve the equations for \(u\) and \(v\), giving both answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. On an Argand diagram, sketch the locus representing complex numbers \(z\) satisfying \(| z + \mathrm { i } | = 1\) and the locus representing complex numbers \(w\) satisfying \(\arg ( w - 2 ) = \frac { 3 } { 4 } \pi\). Find the least value of \(| z - w |\) for points on these loci.

4 Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(| z - 5 - 2 \mathrm { i } | \leqslant \sqrt { 32 }\) and \(\operatorname { Re } ( z ) \geqslant 9\).

2 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - 1 | = 5\) and \(\arg ( z + 4 + 4 \mathrm { i } ) = \frac { 1 } { 4 } \pi\) respectively.

1. Sketch on a single Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Indicate by shading on your Argand diagram the following set of points. $$\{ z : | z - 1 | \leqslant 5 \} \cap \left\{ z : 0 \leqslant \arg ( z + 4 + 4 i ) \leqslant \frac { 1 } { 4 } \pi \right\}$$

5 The complex number \(z\) satisfies the relation $$| z + 4 - 4 i | = 4$$

1. Sketch, on an Argand diagram, the locus of \(z\).
2. Show that the greatest value of \(| z |\) is \(4 ( \sqrt { 2 } + 1 )\).
3. Find the value of \(z\) for which $$\arg ( z + 4 - 4 \mathrm { i } ) = \frac { 1 } { 6 } \pi$$ Give your answer in the form \(a + \mathrm { i } b\).

2

1. Sketch on one diagram:
   1. the locus of points satisfying \(| z - 4 + 2 \mathrm { i } | = 2\);
   2. the locus of points satisfying \(| z | = | z - 3 - 2 \mathrm { i } |\).
2. Shade on your sketch the region in which
   both $$| z - 4 + 2 i | \leqslant 2$$ and $$| z | \leqslant | z - 3 - 2 \mathrm { i } |$$

3 A circle \(C\) and a half-line \(L\) have equations $$| z - 2 \sqrt { 3 } - \mathrm { i } | = 4$$ and $$\arg ( z + i ) = \frac { \pi } { 6 }$$ respectively.

1. Show that:
   1. the circle \(C\) passes through the point where \(z = - \mathrm { i }\);
   2. the half-line \(L\) passes through the centre of \(C\).
2. On one Argand diagram, sketch \(C\) and \(L\).
3. Shade on your sketch the set of points satisfying both $$| z - 2 \sqrt { 3 } - \mathrm { i } | \leqslant 4$$ and $$0 \leqslant \arg ( z + i ) \leqslant \frac { \pi } { 6 }$$

2

1. Indicate on an Argand diagram the region for which \(| z - 4 \mathrm { i } | \leqslant 2\).
2. The complex number \(z\) satisfies \(| z - 4 \mathrm { i } | \leqslant 2\). Find the range of possible values of \(\arg z\).

4

1. On one Argand diagram, sketch the locus of points satisfying:
   1. \(| z - 3 + 2 \mathrm { i } | = 4\);
   2. \(\quad \arg ( z - 1 ) = - \frac { 1 } { 4 } \pi\).
2. Indicate on your sketch the set of points satisfying both $$| z - 3 + 2 i | \leqslant 4$$ and $$\arg ( z - 1 ) = - \frac { 1 } { 4 } \pi$$ (1 mark)

5 The sketch shows an Argand diagram. The points \(A\) and \(B\) represent the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) respectively. The angle \(A O B = 90 ^ { \circ }\) and \(O A = O B\). \includegraphics[max width=\textwidth, alt={}, center]{847295e3-d806-43b1-8d25-688c5558bfe1-3_533_869_852_632}

1. Explain why \(z _ { 2 } = \mathrm { i } z _ { 1 }\).
2. On a single copy of the diagram, draw:
   1. the locus \(L _ { 1 }\) of points satisfying \(\left| z - z _ { 2 } \right| = \left| z - z _ { 1 } \right|\);
   2. the locus \(L _ { 2 }\) of points satisfying \(\arg \left( z - z _ { 2 } \right) = \arg z _ { 1 }\).
3. Find, in terms of \(z _ { 1 }\), the complex number representing the point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).