Intersection of two loci

7

1. Without using a calculator, solve the equation $$3 w + 2 \mathrm { i } w ^ { * } = 17 + 8 \mathrm { i }$$ where \(w ^ { * }\) denotes the complex conjugate of \(w\). Give your answer in the form \(a + b \mathrm { i }\).
2. In an Argand diagram, the loci $$\arg ( z - 2 \mathrm { i } ) = \frac { 1 } { 6 } \pi \quad \text { and } \quad | z - 3 | = | z - 3 \mathrm { i } |$$ intersect at the point \(P\). Express the complex number represented by \(P\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), giving the exact value of \(\theta\) and the value of \(r\) correct to 3 significant figures.

8 The complex number 1 - i is denoted by \(u\).

1. Showing your working and without using a calculator, express $$\frac { \mathrm { i } } { u }$$ in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. On an Argand diagram, sketch the loci representing complex numbers \(z\) satisfying the equations \(| z - u | = | z |\) and \(| z - \mathrm { i } | = 2\).
3. Find the argument of each of the complex numbers represented by the points of intersection of the two loci in part (ii).

7

1. On a single Argand diagram sketch the loci given by the equations \(| z - 3 + 2 i | = 2\) and \(| w - 3 + 2 \mathrm { i } | = | w + 3 - 4 \mathrm { i } |\) where z and \(w\) are complex numbers.
2. Hence find the least value of \(| \mathbf { z } - \mathbf { w } |\) for points on these loci. Give your answer in an exact form.

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

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 \(R\).

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 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by $$| z - 2 \mathrm { i } | = 2 \quad \text { and } \quad | z + 1 | = | z + \mathrm { i } |$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Hence write down the complex numbers represented by the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\). \(7 \quad\) The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r r r } a & 1 & 3 \\ 2 & 1 & - 1 \\ 0 & 1 & 2 \end{array} \right)\).
3. Given that \(\mathbf { B }\) is singular, show that \(a = - \frac { 2 } { 3 }\).
4. Given instead that \(\mathbf { B }\) is non-singular, find the inverse matrix \(\mathbf { B } ^ { - 1 }\).
5. Hence, or otherwise, solve the equations $$\begin{aligned} - x + y + 3 z & = 1 \\ 2 x + y - z & = 4 \\ y + 2 z & = - 1 \end{aligned}$$

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 \(C _ { 2 }\).

5

1. Sketch the locus \(| z - ( 3 + 4 j ) | = 2\) on an Argand diagram.
2. On the same diagram, sketch the locus \(\arg ( z - 4 ) = \frac { 1 } { 2 } \pi\).
3. Indicate clearly on your sketch the points which satisfy both $$| z - ( 3 + 4 j ) | = 2 \quad \text { and } \quad \arg ( z - 4 ) = \frac { 1 } { 2 } \pi$$

2 Indicate on a single Argand diagram

1. the set of points for which \(| z - ( - 3 + 2 \mathrm { j } ) | = 2\),
2. the set of points for which \(\arg ( z - 2 \mathrm { j } ) = \pi\),
3. the two points for which \(| z - ( - 3 + 2 \mathrm { j } ) | = 2\) and \(\arg ( z - 2 \mathrm { j } ) = \pi\).

6 Sketch, on a single Argand diagram, the loci given by \(| z - \sqrt { 3 } - \mathrm { i } | = 2\) and \(\arg z = \frac { 1 } { 6 } \pi\).

5 The complex number \(1 + \mathrm { i } \sqrt { 3 }\) is denoted by \(a\).

1. Find \(| a |\) and \(\arg a\).
2. Sketch on a single Argand diagram the loci given by \(| z - a | = | a |\) and \(\arg ( z - a ) = \frac { 1 } { 2 } \pi\).

7 In this question you must show detailed reasoning.
Two loci, \(C _ { 1 }\) and \(C _ { 2 }\), are defined as follows. \(\mathrm { C } _ { 1 } = \left\{ \mathrm { z } : \arg ( \mathrm { z } + 2 - \mathrm { i } ) = \frac { 1 } { 4 } \pi \right\}\) and \(\mathrm { C } _ { 2 } = \left\{ \mathrm { z } : \arg ( \mathrm { z } - 2 - \sqrt { 3 } - 2 \mathrm { i } ) = \frac { 2 } { 3 } \pi \right\}\) By considering the representations of \(C _ { 1 }\) and \(C _ { 2 }\) on an Argand diagram, determine the locus \(C _ { 1 } \cap C _ { 2 }\).

4 The Argand diagram shows a circle of radius 3. The centre of the circle is the point which represents the complex number \(4 - 2 \mathrm { i }\). \includegraphics[max width=\textwidth, alt={}, center]{4159328b-475e-4f29-91f2-f2f343573251-3_417_775_349_644}

1. Use set notation to define the locus of complex numbers, \(z\), represented by points which lie on the circle. The locus \(L\) is defined by \(\mathrm { L } = \{ \mathrm { z } : \mathrm { z } \in \mathbb { C } , | \mathrm { z } - \mathrm { i } | = | \mathrm { z } + 2 | \}\).
2. On the Argand diagram in the Printed Answer Booklet, sketch and label the locus \(L\). You are given that the locus \(\left\{ z : z \in \mathbb { C } , \arg ( z - 1 ) = \frac { 1 } { 4 } \pi , \operatorname { Re } ( z ) = 3 \right\}\) contains only one number.
3. Find this number.

8 Two loci, \(C _ { 1 }\) and \(C _ { 2 }\), are defined by $$\begin{aligned} & C _ { 1 } = \left\{ z : | z | = \left| z - 4 d ^ { 2 } - 36 \right| \right\} \\ & C _ { 2 } = \left\{ z : \arg ( z - 12 d - 3 i ) = \frac { 1 } { 4 } \pi \right\} \end{aligned}$$ where \(d\) is a real number.

1. Find, in terms of \(d\), the complex number which is represented on an Argand diagram by the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
   [0pt] [You may assume that \(C _ { 1 } \cap C _ { 2 } \neq \varnothing\).]
2. Explain why the solution found in part (a) is not valid when \(d = 3\). \section*{END OF QUESTION PAPER} \section*{OCR
   Oxford Cambridge and RSA}

1

1. Sketch on a single Argand diagram the loci given by
   1. \(\quad | z - 1 + 2 i | = 3\),
   2. \(\quad | z + 1 | = | z - 2 |\).
2. Indicate, by shading, the region of the Argand diagram for which \(| z - 1 + 2 i | \leqslant 3\) and \(| z + 1 | \leqslant | z - 2 |\).

2

1. On the same Argand diagram, draw:
   1. the locus of points satisfying \(| z - 4 + 2 \mathrm { i } | = 4\);
   2. the locus of points satisfying \(| z | = | z - 2 \mathrm { i } |\).
2. Indicate on your sketch the set of points satisfying both $$| z - 4 + 2 i | \leqslant 4$$ and $$| z | \geqslant | z - 2 \mathrm { i } |$$

2 Two loci, \(L _ { 1 }\) and \(L _ { 2 }\), in an Argand diagram are given by $$\begin{aligned} & L _ { 1 } : | z + 6 - 5 \mathrm { i } | = 4 \sqrt { 2 } \\ & L _ { 2 } : \quad \arg ( z + \mathrm { i } ) = \frac { 3 \pi } { 4 } \end{aligned}$$ The point \(P\) represents the complex number \(- 2 + \mathrm { i }\).

1. Verify that the point \(P\) is a point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).
2. Sketch \(L _ { 1 }\) and \(L _ { 2 }\) on one Argand diagram.
3. The point \(Q\) is also a point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\). Find the complex number that is represented by \(Q\).

4

1. A circle \(C\) in the Argand diagram has equation $$| z + 5 - \mathrm { i } | = \sqrt { 2 }$$ Write down its radius and the complex number representing its centre.
2. A half-line \(L\) in the Argand diagram has equation $$\arg ( z + 2 \mathrm { i } ) = \frac { 3 \pi } { 4 }$$ Show that \(z _ { 1 } = - 4 + 2 \mathrm { i }\) lies on \(L\).
3. 1. Show that \(z _ { 1 } = - 4 + 2 \mathrm { i }\) also lies on \(C\).
   2. Hence show that \(L\) touches \(C\).
   3. Sketch \(L\) and \(C\) on one Argand diagram.
4. The complex number \(z _ { 2 }\) lies on \(C\) and is such that \(\arg \left( z _ { 2 } + 2 \mathrm { i } \right)\) has as great a value as possible. Indicate the position of \(z _ { 2 }\) on your sketch.

3 Two loci, \(L _ { 1 }\) and \(L _ { 2 }\), in an Argand diagram are given by $$\begin{aligned} & L _ { 1 } : | z + 1 + 3 \mathrm { i } | = | z - 5 - 7 \mathrm { i } | \\ & L _ { 2 } : \arg z = \frac { \pi } { 4 } \end{aligned}$$

1. Verify that the point represented by the complex number \(2 + 2 \mathrm { i }\) is a point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).
2. Sketch \(L _ { 1 }\) and \(L _ { 2 }\) on one Argand diagram.
3. Shade on your Argand diagram the region satisfying
   both $$| z + 1 + 3 i | \leqslant | z - 5 - 7 i |$$ and $$\frac { \pi } { 4 } \leqslant \arg z \leqslant \frac { \pi } { 2 }$$

1

1. Draw on the same Argand diagram:
   1. the locus of points for which $$| z - 2 - 5 \mathrm { i } | = 5$$
   2. the locus of points for which $$\arg ( z + 2 i ) = \frac { \pi } { 4 }$$
2. Indicate on your diagram the set of points satisfying both $$| z - 2 - 5 i | \leqslant 5$$ and $$\arg ( z + 2 \mathrm { i } ) = \frac { \pi } { 4 }$$ (2 marks)

2

1. Draw on the Argand diagram below:
   1. the locus of points for which $$| z - 2 - 3 \mathrm { i } | = 2$$
   2. the locus of points for which $$| z + 2 - \mathrm { i } | = | z - 2 |$$
2. Indicate on your diagram the points satisfying both $$| z - 2 - 3 \mathrm { i } | = 2$$ and $$| z + 2 - \mathrm { i } | \leqslant | z - 2 |$$ (l mark) \includegraphics[max width=\textwidth, alt={}, center]{ff63460d-0fa1-437d-bc08-3e7ce809e32b-3_1404_1431_1043_319}

7

1. Sketch on a single Argand diagram
   1. the set of points for which \(| z - 1 - 3 i | = 3\),
   2. the set of points for which \(\arg ( z + 4 ) = \frac { 1 } { 4 } \pi\).
2. Find, in exact form, the two values of \(z\) for which \(| z - 1 - 3 i | = 3\) and \(\arg ( z + 4 ) = \frac { 1 } { 4 } \pi\).

8 In an Argand diagram, the point P representing the complex number \(w\) lies on the locus defined by \(\left\{ z : \arg ( z - 7 ) = \frac { 3 } { 4 } \pi \right\}\). You are given that \(\operatorname { Re } ( w ) = 1\).

1. Find \(w\). The point P also lies on the locus defined by \(\{ \mathrm { z } : | \mathrm { z } + 3 - 9 \mathrm { i } | = \mathrm { k } \}\), where \(k\) is a constant.
2. Find the complex number represented by the other point of intersection of the loci defined by $$\{ z : | z + 3 - 9 i | = k \} \text { and } \left\{ z : \arg ( z - 7 ) = \frac { 3 } { 4 } \pi \right\} .$$