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.

7 questions · Standard +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 MEI FP1 2007 June Q2
3 marks Moderate -0.8
2 Write down the equation of the locus represented by the circle in the Argand diagram shown in Fig. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d7e20cfb-da04-4d7b-bcda-53f99f6faec4-2_581_600_872_737} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
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 Further Pure Core Specimen Q2
6 marks Standard +0.8
2
  1. On an Argand diagram draw the locus of points which satisfy \(\arg ( z - 4 \mathrm { i } ) = \frac { \pi } { 4 }\).
  2. Give, in complex form, the equation of the circle which has centre at \(6 + 4 \mathrm { i }\) and touches the locus in part (i).