Circle equations in complex form

A question is this type if and only if it asks to find or verify the equation of a circle in the form |z - a| = r, including finding center and radius from given conditions.

13 questions · Moderate -0.1

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CAIE P3 2015 June Q8
9 marks Standard +0.3
8 The complex number \(w\) is defined by \(w = \frac { 22 + 4 \mathrm { i } } { ( 2 - \mathrm { i } ) ^ { 2 } }\).
  1. Without using a calculator, show that \(w = 2 + 4 \mathrm { i }\).
  2. It is given that \(p\) is a real number such that \(\frac { 1 } { 4 } \pi \leqslant \arg ( w + p ) \leqslant \frac { 3 } { 4 } \pi\). Find the set of possible values of \(p\).
  3. The complex conjugate of \(w\) is denoted by \(w ^ { * }\). The complex numbers \(w\) and \(w ^ { * }\) are represented in an Argand diagram by the points \(S\) and \(T\) respectively. Find, in the form \(| z - a | = k\), the equation of the circle passing through \(S , T\) and the origin.
CAIE P3 2011 November Q6
8 marks Standard +0.3
6 The complex number \(w\) is defined by \(w = - 1 + \mathrm { i }\).
  1. Find the modulus and argument of \(w ^ { 2 }\) and \(w ^ { 3 }\), showing your working.
  2. The points in an Argand diagram representing \(w\) and \(w ^ { 2 }\) are the ends of a diameter of a circle. Find the equation of the circle, giving your answer in the form \(| z - ( a + b \mathrm { i } ) | = k\).
OCR FP1 Specimen Q6
10 marks Standard +0.3
6 In an Argand diagram, the variable point \(P\) represents the complex number \(z = x + \mathrm { i } y\), and the fixed point \(A\) represents \(a = 4 - 3 \mathrm { i }\).
  1. Sketch an Argand diagram showing the position of \(A\), and find \(| a |\) and \(\arg a\).
  2. Given that \(| z - a | = | a |\), sketch the locus of \(P\) on your Argand diagram.
  3. Hence write down the non-zero value of \(z\) corresponding to a point on the locus for which
    1. the real part of \(z\) is zero,
    2. \(\quad \arg z = \arg a\).
OCR FP1 2016 June Q6
9 marks Standard +0.3
6 In an Argand diagram the points \(A\) and \(B\) represent the complex numbers \(5 + 4 \mathrm { i }\) and \(1 + 2 \mathrm { i }\) respectively.
  1. Given that \(A\) and \(B\) are the ends of a diameter of a circle \(C\), find the equation of \(C\) in complex number form. The perpendicular bisector of \(A B\) is denoted by \(l\).
  2. Sketch \(C\) and \(l\) on a single Argand diagram.
  3. Find the complex numbers represented by the points of intersection of \(C\) and \(l\).
OCR MEI FP1 2009 January Q4
3 marks Moderate -0.8
4 Write down the equation of the locus represented in the Argand diagram shown in Fig. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{35094899-149c-438e-b6c8-b333d2fefc0c-2_474_497_1932_824} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
AQA FP2 2014 June Q2
8 marks Standard +0.8
2
  1. Sketch, on the Argand diagram below, the locus \(L\) of points satisfying $$\arg ( z - 2 \mathrm { i } ) = \frac { 2 \pi } { 3 }$$
    1. A circle \(C\), of radius 3, has its centre lying on \(L\) and touches the line \(\operatorname { Im } ( z ) = 2\). Sketch \(C\) on the Argand diagram used in part (a).
    2. Find the centre of \(C\), giving your answer in the form \(a + b \mathrm { i }\).
      [0pt] [3 marks]
OCR MEI FP1 2007 June Q2
3 marks Easy -1.2
Write down the equation of the locus represented by the circle in the Argand diagram shown in Fig. 2. [3] \includegraphics{figure_2}
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]
AQA Further Paper 1 2023 June Q2
1 marks Easy -1.2
The diagram below shows a locus on an Argand diagram. \includegraphics{figure_2} Which of the equations below represents the locus shown above? Circle your answer. [1 mark] \(|z - 2 + 3\mathrm{i}| = 2 \quad |z + 2 - 3\mathrm{i}| = 2 \quad |z - 2 + 3\mathrm{i}| = 4 \quad |z + 2 - 3\mathrm{i}| = 4\)
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]
SPS SPS FM Pure 2025 February Q5
9 marks Standard +0.3
In an Argand diagram, the points \(A\) and \(B\) are represented by the complex numbers \(-3 + 2i\) and \(5 - 4i\) respectively. The points \(A\) and \(B\) are the end points of a diameter of a circle \(C\).
  1. Find the equation of \(C\), giving your answer in the form $$|z - a| = b \quad a \in \mathbb{C}, \quad b \in \mathbb{R}$$ [3]
The circle \(D\), with equation \(|z - 2 - 3i| = 2\), intersects \(C\) at the points representing the complex numbers \(z_1\) and \(z_2\).
  1. Find the complex numbers \(z_1\) and \(z_2\). [6]
SPS SPS FM Pure 2025 February Q5
9 marks Standard +0.3
In an Argand diagram, the points \(A\) and \(B\) are represented by the complex numbers \(-3 + 2i\) and \(5 - 4i\) respectively. The points \(A\) and \(B\) are the end points of a diameter of a circle \(C\).
  1. Find the equation of \(C\), giving your answer in the form $$|z - a| = b \quad a \in \mathbb{C}, \, b \in \mathbb{R}$$ [3]
The circle \(D\), with equation \(|z - 2 - 3i| = 2\), intersects \(C\) at the points representing the complex numbers \(z_1\) and \(z_2\)
  1. Find the complex numbers \(z_1\) and \(z_2\) [6]