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

Edexcel FP2 Q22

10 marks Standard +0.8

1. 1. On the same Argand diagram sketch the loci given by the following equations. $$|z - 1| = 1,$$ $$\arg(z + 1) = \frac{\pi}{12},$$ $$\arg(z + 1) = \frac{\pi}{2}.$$ [4]
   2. Shade on your diagram the region for which $$|z - 1| \leq 1 \quad \text{and} \quad \frac{\pi}{12} \leq \arg(z + 1) \leq \frac{\pi}{2}.$$ [1]
2. 1. Show that the transformation $$w = \frac{z - 1}{z}, \quad z \neq 0,$$ maps \(|z - 1| = 1\) in the \(z\)-plane onto \(|w| = |w - 1|\) in the \(w\)-plane. [3] The region \(|z - 1| \leq 1\) in the \(z\)-plane is mapped onto the region \(T\) in the \(w\)-plane.
   2. Shade the region \(T\) on an Argand diagram. [2]

OCR FP1 Q6

7 marks Standard +0.3

The loci \(C_1\) and \(C_2\) are given by $$|z - 2i| = 2 \quad \text{and} \quad |z + 1| = |z + i|$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
2. Hence write down the complex numbers represented by the points of intersection of \(C_1\) and \(C_2\). [2]

OCR FP1 2013 January Q7

7 marks Moderate -0.3

1. Sketch on a single Argand diagram the loci given by
   1. \(|z| = 2\), [2]
   2. \(\arg(z - 3 - i) = \pi\). [3]
2. Indicate, by shading, the region of the Argand diagram for which $$|z| < 2 \text{ and } 0 < \arg(z - 3 - i) < \pi.$$ [2]

OCR FP1 2005 June Q6

7 marks Moderate -0.3

The loci \(C_1\) and \(C_2\) are given by $$|z - 2\text{i}| = 2 \quad \text{and} \quad |z + 1| = |z + \text{i}|$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
2. Hence write down the complex numbers represented by the points of intersection of \(C_1\) and \(C_2\). [2]

OCR FP1 2010 June Q6

6 marks Moderate -0.3

1. Sketch on a single Argand diagram the loci given by
   1. \(|z - 3 + 4\text{i}| = 5\), [2]
   2. \(|z| = |z - 6|\). [2]
2. Indicate, by shading, the region of the Argand diagram for which $$|z - 3 + 4\text{i}| \leq 5 \quad \text{and} \quad |z| \geq |z - 6|.$$ [2]

OCR MEI FP1 2006 June Q4

8 marks Moderate -0.8

Indicate, on separate Argand diagrams,

1. the set of points \(z\) for which \(|z-(3-\mathrm{j})| \leqslant 3\), [3]
2. the set of points \(z\) for which \(1 < |z-(3-\mathrm{j})| \leqslant 3\), [2]
3. the set of points \(z\) for which \(\arg(z-(3-\mathrm{j})) = \frac{1}{4}\pi\). [3]

AQA FP2 2013 January Q2

10 marks Standard +0.3

Two loci, \(L_1\) and \(L_2\), in an Argand diagram are given by $$L_1 : |z + 6 - 5\text{i}| = 4\sqrt{2}$$ $$L_2 : \arg(z + \text{i}) = \frac{3\pi}{4}$$ The point \(P\) represents the complex number \(-2 + \text{i}\).

1. Verify that the point \(P\) is a point of intersection of \(L_1\) and \(L_2\). [2 marks]
2. Sketch \(L_1\) and \(L_2\) on one Argand diagram. [6 marks]
3. The point \(Q\) is also a point of intersection of \(L_1\) and \(L_2\). Find the complex number that is represented by \(Q\). [2 marks]

AQA FP2 2011 June Q1

8 marks Moderate -0.3

1. Draw on the same Argand diagram:
   1. the locus of points for which $$|z - 2 - 5i| = 5$$ [3 marks]
   2. the locus of points for which $$\arg(z + 2i) = \frac{\pi}{4}$$ [3 marks]
2. Indicate on your diagram the set of points satisfying both $$|z - 2 - 5i| \leqslant 5$$ and $$\arg(z + 2i) = \frac{\pi}{4}$$ [2 marks]

OCR FP3 Q1

3 marks Easy -1.2

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

OCR FP3 Q4

9 marks Standard +0.8

1. By expressing \(\cos \theta\) in terms of \(e^{i\theta}\) and \(e^{-i\theta}\), show that $$\cos^5 \theta \equiv \frac{1}{16}(\cos 5\theta + 5\cos 3\theta + 10\cos \theta).$$ [5]
2. Hence solve the equation \(\cos 5\theta + 5\cos 3\theta + 9\cos \theta = 0\) for \(0 \leqslant \theta \leqslant \pi\). [4]

AQA Further AS Paper 1 2018 June Q14

7 marks Challenging +1.2

1. Sketch, on the Argand diagram below, the locus of points satisfying the equation $$|z - 3| = 2$$ [1 mark] \includegraphics{figure_14a}
2. There is a unique complex number \(w\) that satisfies both $$|w - 3| = 2 \quad \text{and} \quad \arg(w + 1) = \alpha$$ where \(\alpha\) is a constant such that \(0 < \alpha < \pi\)
   1. Find the value of \(\alpha\). [2 marks]
   2. Express \(w\) in the form \(r(\cos \theta + i \sin \theta)\). Give each of \(r\) and \(\theta\) to two significant figures. [4 marks]

AQA Further AS Paper 1 2020 June Q18

5 marks Standard +0.8

The locus of points \(L_1\) satisfies the equation \(|z| = 2\) The locus of points \(L_2\) satisfies the equation \(\arg(z + 4) = \frac{\pi}{4}\)

1. Sketch \(L_1\) on the Argand diagram below. \includegraphics{figure_18} [1 mark]
2. Sketch \(L_2\) on the Argand diagram above. [1 mark]
3. The complex number \(a + ib\), where \(a\) and \(b\) are real, lies on \(L_1\) The complex number \(c + id\), where \(c\) and \(d\) are real, lies on \(L_2\) Calculate the least possible value of the expression $$(c - a)^2 + (d - b)^2$$ [3 marks]

AQA Further Paper 1 2022 June Q8

11 marks Standard +0.8

1. The complex number \(w\) is such that $$\arg(w + 2i) = \tan^{-1}\frac{1}{2}$$ It is given that \(w = x + iy\), where \(x\) and \(y\) are real and \(x > 0\) Find an equation for \(y\) in terms of \(x\) [2 marks]
2. The complex number \(z\) satisfies both $$-\frac{\pi}{2} \leq \arg(z + 2i) \leq \tan^{-1}\frac{1}{2} \quad \text{and} \quad |z - 2 + 3i| \leq 2$$ The region \(R\) is the locus of \(z\) Sketch the region \(R\) on the Argand diagram below. [4 marks] \includegraphics{figure_1}
3. \(z_1\) is the point in \(R\) at which \(|z|\) is minimum.
   1. Calculate the exact value of \(|z_1|\) [3 marks]
   2. Express \(z_1\) in the form \(a + ib\), where \(a\) and \(b\) are real. [2 marks]

AQA Further Paper 2 2019 June Q6

6 marks Challenging +1.8

A circle \(C\) in the complex plane has equation \(|z - 2 - 5\mathrm{i}| = a\) The point \(z_1\) on \(C\) has the least argument of any point on \(C\), and \(\arg(z_1) = \frac{\pi}{4}\) Prove that \(a = \frac{3\sqrt{2}}{2}\) [6 marks]

AQA Further Paper 2 2024 June Q17

9 marks Standard +0.8

The Argand diagram below shows a circle \(C\) \includegraphics{figure_17}

1. Write down the equation of the locus of \(C\) in the form $$|z - w| = a$$ where \(w\) is a complex number whose real and imaginary parts are integers, and \(a\) is an integer. [2 marks]
2. It is given that \(z_1\) is a complex number representing a point on \(C\). Of all the complex numbers which represent points on \(C\), \(z_1\) has the least argument.
   1. Find \(|z_1|\) Give your answer in an exact form. [3 marks]
   2. Show that \(\arg z_1 = \arcsin\left(\frac{6\sqrt{3} - 2}{13}\right)\) [4 marks]

OCR Further Pure Core AS 2020 November Q8

8 marks Challenging +1.8

Two loci, \(C_1\) and \(C_2\), are defined by $$C_1 = \{z:|z| = |z - 4d^2 - 36|\}$$ $$C_2 = \left\{z:\arg(z - 12d - 3\text{i}) = \frac{1}{4}\pi\right\}$$ 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 \(C_2\). [You may assume that \(C_1 \cap C_2 \neq \emptyset\).] [6]
2. Explain why the solution found in part (a) is not valid when \(d = 3\). [2]

OCR Further Pure Core 1 2021 November Q1

6 marks Moderate -0.8

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

OCR MEI Further Pure Core AS 2018 June Q9

9 marks Standard +0.3

Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\left\{z : |z| \leq 4\sqrt{2}\right\} \cap \left\{z : -\frac{1}{4}\pi \leq \arg z \leq \frac{1}{4}\pi\right\}.$$ \includegraphics{figure_9}

1. Find, in modulus-argument form, the complex number represented by the point P. [2]
2. Find, in the form \(a + ib\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
   lie within this region. [4]

OCR MEI Further Pure Core AS Specimen Q3

4 marks Moderate -0.3

1. Write down, in complex form, the equation of the locus represented by the circle in the Argand diagram shown in Fig. 3. [2] \includegraphics{figure_3}
2. On the copy of Fig. 3 in the Printed Answer Booklet mark with a cross any point(s) on the circle for which \(\arg(z - 2i) = \frac{\pi}{4}\). [2]

OCR MEI Further Pure Core Specimen Q2

6 marks Standard +0.8

1. On an Argand diagram draw the locus of points which satisfy \(\arg(z - 4i) = \frac{\pi}{4}\). [2]
2. Give, in complex form, the equation of the circle which has centre at \(6 + 4i\) and touches the locus in part (i). [4]

WJEC Further Unit 1 2018 June Q7

5 marks Standard +0.3

The complex number \(z\) is represented by the point \(P(x, y)\) in the Argand diagram and $$|z - 4 - \mathrm{i}| = |z + 2|.$$

1. Find the equation of the locus of \(P\). [4]
2. Give a geometric interpretation of the locus of \(P\). [1]

WJEC Further Unit 1 Specimen Q5

9 marks Standard +0.8

The complex number \(z\) is represented by the point \(P(x, y)\) in an Argand diagram and $$|z - 3| = 2|z + i|.$$ Show that the locus of \(P\) is a circle and determine its radius and the coordinates of its centre. [9]

SPS SPS FM 2020 December Q9

5 marks Moderate -0.3

Sketch on an Argand diagram the locus of all points that satisfy \(|z + 4 - 4i| = 2\sqrt{2}\) and hence find \(\theta, \phi \in (-\pi, \pi]\) such that \(\theta \leq \arg z \leq \phi\). [5]

SPS SPS FM Pure 2021 June Q16

7 marks Challenging +1.8

Given that there are two distinct complex numbers \(z\) that satisfy $$\{z: |z - 3 - 5i| = 2r\} \cap \left\{z: \arg(z - 2) = \frac{3\pi}{4}\right\}$$ determine the exact range of values for the real constant \(r\). [7]

SPS SPS FM Pure 2021 May Q6

6 marks Challenging +1.2

A circle \(C\) in the complex plane has equation \(|z - 2 - 5i| = a\). The point \(z_1\) on \(C\) has the least argument of any point on \(C\), and \(arg(z_1) = \frac{\pi}{4}\). Prove that \(a = \frac{3\sqrt{2}}{2}\). [6]