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

6

1. Two points, \(A\) and \(B\), on an Argand diagram are represented by the complex numbers \(2 + 3 \mathrm { i }\) and \(- 4 - 5 \mathrm { i }\) respectively. Given that the points \(A\) and \(B\) are at the ends of a diameter of a circle \(C _ { 1 }\), express the equation of \(C _ { 1 }\) in the form \(\left| z - z _ { 0 } \right| = k\).
2. A second circle, \(C _ { 2 }\), is represented on the Argand diagram by the equation \(| z - 5 + 4 \mathrm { i } | = 4\). Sketch on one Argand diagram both \(C _ { 1 }\) and \(C _ { 2 }\).
3. The points representing the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) lie on \(C _ { 1 }\) and \(C _ { 2 }\) respectively and are such that \(\left| z _ { 1 } - z _ { 2 } \right|\) has its maximum value. Find this maximum value, giving your answer in the form \(a + b \sqrt { 5 }\).

5 The locus of points, \(L\), satisfies the equation $$| z - 2 + 4 \mathrm { i } | = | z |$$

1. Sketch \(L\) on the Argand diagram below.
2. The locus \(L\) cuts the real axis at \(A\) and the imaginary axis at \(B\).
   1. Show that the complex number represented by \(C\), the midpoint of \(A B\), is $$\frac { 5 } { 2 } - \frac { 5 } { 4 } \mathrm { i }$$
   2. The point \(O\) is the origin of the Argand diagram. Find the equation of the circle that passes through the points \(O , A\) and \(B\), giving your answer in the form \(| z - \alpha | = k\).
      [0pt] [2 marks] \section*{(a)}
      \includegraphics[max width=\textwidth, alt={}]{bc3aaed2-4aef-4aec-b657-098b1e581e55-10_1173_1242_1217_463}

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 $$\arg ( z + 4 - 4 i ) = \frac { 1 } { 6 } \pi$$ Give your answer in the form \(a + \mathrm { i } b\).

12

1. Sketch, on the Argand diagram below, the locus of points satisfying the equation $$| z - 2 \mathrm { i } | = 2$$
   \includegraphics[max width=\textwidth, alt={}]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-18_1219_1260_477_392}
   12
2. Sketch, also on the Argand diagram above, the locus of points satisfying the equation $$\arg z = \frac { \pi } { 3 }$$ [1 mark] 12
3. For the complex number \(w\) find the maximum value of \(| w |\) such that $$| w - 2 \mathrm { i } | \leq 2 \quad \text { and } \quad 0 \leq \arg w \leq \frac { \pi } { 3 }$$ $$y = \frac { 2 x + 7 } { 3 x + 5 }$$

17 The circle \(C\) represents the locus of points satisfying the equation $$| z - a \mathrm { i } | = b$$ where \(a\) and \(b\) are real constants. The circle \(C\) intersects the imaginary axis at 2 i and 8 i
The circle \(C\) is shown on the Argand diagram in Figure 2 \begin{figure}[h] \end{figure} 17

1. 1. Write down the value of \(a\) 17
      1. (ii) Write down the value of \(b\) 17
   2. The half-line \(L\) represents the locus of points satisfying the equation $$\arg ( z ) = \tan ^ { - 1 } ( k )$$ where \(k\) is a positive constant.
      The point \(P\) is the only point which lies on both \(C\) and \(L\), as shown in Figure 3 \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-25_766_770_685_699}
      \end{figure} 17
   3. 1. The point \(O\) represents the number \(0 + 0 \mathrm { i }\) Calculate the length \(O P\) 17
   4. (ii) Calculate the exact value of \(k\) 17
   5. (iii) Find the complex number represented by point \(P\) Give your answer in the form \(x + y i\) where \(x\) and \(y\) are real.

8 The complex number \(z\) satisfies the equations $$\left| z ^ { * } - 1 - 2 i \right| = | z - 3 |$$ and $$| z - a | = 3$$ where \(a\) is real.
Show that \(a\) must lie in the interval \([ 1 - s \sqrt { t } , 1 + s \sqrt { t } ]\), where \(s\) and \(t\) are prime numbers.
[0pt] [6 marks]

4

1. A locus \(C _ { 1 }\) is defined by \(C _ { 1 } = \{ \mathrm { z } : | \mathrm { z } + \mathrm { i } | \leqslant \mid \mathrm { z } - 2 \}\).
   1. Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing \(C _ { 1 }\).
   2. Find the cartesian equation of the boundary line of the region representing \(C _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\).
2. A locus \(C _ { 2 }\) is defined by \(C _ { 2 } = \{ \mathrm { z } : | \mathrm { z } + 1 | \leqslant 3 \} \cap \{ \mathrm { z } : | \mathrm { z } - 2 \mathrm { i } | \geqslant 2 \}\). Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing \(C _ { 2 }\).

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

3

1. On a single copy of an Argand diagram, sketch the loci defined by $$| z + 2 | = 3 \quad \text { and } \quad \arg ( z - \mathrm { i } ) = - \frac { 1 } { 4 } \pi$$
2. State the complex number \(z\) which corresponds to the point of intersection of these two loci.

5 The complex numbers \(u\) and \(v\) are given by \(u = 3 + 2 \mathrm { i }\) and \(v = 1 + 4 \mathrm { i }\).

1. Given that \(a u ^ { 2 } + b v ^ { * } = 7 + 36 \mathrm { i }\) find the values of the real constants \(a\) and \(b\).
2. Show the points representing \(u\) and \(v\) on an Argand diagram and hence sketch the locus given by \(| z - u | = | z - v |\). Find the point of intersection of this locus with the imaginary axis.

2

1. On a single Argand diagram, sketch and clearly label each of the following loci:
   1. \(| z | = 4\),
   2. \(\quad \arg ( z + 4 \mathrm { i } ) = \frac { 1 } { 4 } \pi\).
   3. On the same Argand diagram, shade the region \(R\) defined by the inequalities $$| z | \leqslant 4 \quad \text { and } \quad 0 \leqslant \arg ( z + 4 i ) \leqslant \frac { 1 } { 4 } \pi$$

Throughout this question the use of a calculator is not permitted. The complex number \(2 - \mathrm{i}\) is denoted by \(u\).

1. It is given that \(u\) is a root of the equation \(x^3 + ax^2 - 3x + b = 0\), where the constants \(a\) and \(b\) are real. Find the values of \(a\) and \(b\). [4]
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(|z - u| < 1\) and \(|z| < |z + \mathrm{i}|\). [4]

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

1. The complex numbers \(u\) and \(v\) satisfy the equations $$u + 2v = 2i \quad \text{and} \quad iu + v = 3.$$ Solve the equations for \(u\) and \(v\), giving both answers in the form \(x + iy\), where \(x\) and \(y\) are real. [5]
2. On an Argand diagram, sketch the locus representing complex numbers \(z\) satisfying \(|z + i| = 1\) and the locus representing complex numbers \(w\) satisfying \(\arg(w - 2) = \frac{\pi}{4}\). Find the least value of \(|z - w|\) for points on these loci. [5]

Given that \(z = e^{i\theta}\) and \(n\) is a positive integer, show that $$z^n + \frac{1}{z^n} = 2 \cos n\theta \quad \text{and} \quad z^n - \frac{1}{z^n} = 2i \sin n\theta.$$ [2] Hence express \(\sin^6 \theta\) in the form $$p \cos 6\theta + q \cos 4\theta + r \cos 2\theta + s,$$ where the constants \(p\), \(q\), \(r\), \(s\) are to be determined. [4] Hence find the mean value of \(\sin^6 \theta\) with respect to \(\theta\) over the interval \(0 \leq \theta \leq \frac{1}{4}\pi\). [5]

1. State the sum of the series \(1 + z + z^2 + \ldots + z^{n-1}\), for \(z \neq 1\). [1]
2. By letting \(z = \cos\theta + i\sin\theta\), where \(\cos\theta \neq 1\), show that $$1 + \cos\theta + \cos 2\theta + \ldots + \cos(n-1)\theta = \frac{1}{2}\left(1 - \cos n\theta + \frac{\sin n\theta \sin\theta}{1 - \cos\theta}\right).$$ [7]

\includegraphics{figure_8} The diagram shows the curve with equation \(y = \cos x\) for \(0 \leq x \leq 1\), together with a set of \(n\) rectangles of width \(\frac{1}{n}\).

1. By considering the sum of the areas of these rectangles, show that $$\int_0^1 \cos x\,dx < \frac{1}{2n}\left(1 - \cos 1 + \frac{\sin 1\sin\frac{1}{n}}{1 - \cos\frac{1}{n}}\right).$$ [4]
2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int_0^1 \cos x\,dx\). [3]

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]
2. Find the complex numbers \(z\) which satisfy both \(|z - 6| = |z|\) and \(|z - 3 - 4i| = 5\). [3]

The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \(w = \frac{30}{z}\).

1. Show that \(T\) maps \(|z - 6| = |z|\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle. [5]

The point \(P\) represents the complex number \(z\) on an Argand diagram, where $$|z - i| = 2.$$ The locus of \(P\) as \(z\) varies is the curve \(C\).

1. Find a cartesian equation of \(C\). [2]
2. Sketch the curve \(C\). [2]

A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{z + i}{3 + iz}, \quad z \neq 3i.$$ The point \(Q\) is mapped by \(T\) onto the point \(R\). Given that \(R\) lies on the real axis,

1. show that \(Q\) lies on \(C\). [5]

1. Express \(\frac{1}{r(r + 2)}\) in partial fractions. [2]
2. Hence prove, by the method of differences, that $$\sum_{r=1}^{2n} \frac{1}{r(r + 2)} = \frac{n(4n + 5)}{4(n + 1)(n + 2)},$$ [6]

The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$|z - 6| = 2|z - 3|.$$

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. [6]

where \(a\) and \(b\) are constants to be found.

1. Hence show that $$\sum_{r=1}^{2n} \frac{1}{r(r + 2)} = \frac{n(4n + 5)}{4(n + 1)(n + 2)},$$ [3]
2. Find the complex number for which both \(|z - 6| = 2|z - 3|\) and \(\arg(z - 6) = -\frac{3\pi}{4}\). [4]

The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$|z - 3| = 2|z|.$$

1. Show that, as \(z\) varies, the locus of \(P\) is a circle, and give the coordinates of the centre and the radius of the circle.(5)

The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$|z + 3| = |z - i\sqrt{3}|.$$

1. Sketch, on the same Argand diagram, the locus of \(P\) and the locus of \(Q\) as \(z\) varies.(5)
2. On your diagram shade the region which satisfies $$|z - 3| \geq 2|z| \text{ and } |z + 3| \geq |z - i\sqrt{3}|.$$ (2)