Cartesian equation from modulus equality

A question is this type if and only if it asks to derive the Cartesian equation from a locus condition of the form |z - a| = |z - b| (perpendicular bisector) or |z - a| = k|z - b| where k ≠ 1 (circle of Apollonius).

13 questions · Standard +0.2

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CAIE P3 2013 June Q7
8 marks Standard +0.3
7 The complex number \(z\) is defined by \(z = a + \mathrm { i } b\), where \(a\) and \(b\) are real. The complex conjugate of \(z\) is denoted by \(z ^ { * }\).
  1. Show that \(| z | ^ { 2 } = z z ^ { * }\) and that \(( z - k \mathrm { i } ) ^ { * } = z ^ { * } + k \mathrm { i }\), where \(k\) is real. In an Argand diagram a set of points representing complex numbers \(z\) is defined by the equation \(| z - 10 \mathrm { i } | = 2 | z - 4 \mathrm { i } |\).
  2. Show, by squaring both sides, that $$z z ^ { * } - 2 \mathrm { i } z ^ { * } + 2 \mathrm { i } z - 12 = 0$$ Hence show that \(| z - 2 i | = 4\).
  3. Describe the set of points geometrically.
Edexcel F2 2024 June Q1
4 marks Moderate -0.3
  1. The complex number \(z = x + i y\) satisfies the equation
$$| z - 3 - 4 i | = | z + 1 + i |$$
  1. Determine an equation for the locus of \(z\) giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
  2. Shade, on an Argand diagram, the region defined by $$| z - 3 - 4 i | \leqslant | z + 1 + i |$$ You do not need to determine the coordinates of any intercepts on the coordinate axes.
WJEC Further Unit 1 2018 June Q7
5 marks Moderate -0.5
7. 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 \(P\).
  2. Give a geometric interpretation of the locus of \(P\).
WJEC Further Unit 1 2019 June Q6
3 marks Moderate -0.5
6. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram. Given that $$| z - 1 | = | z - 2 \mathrm { i } |$$ show that the locus of \(P\) is a straight line.
WJEC Further Unit 1 2022 June Q5
4 marks Moderate -0.5
5. (a) The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram. Given that $$| z - 3 + 2 i | = | z - 3 |$$ find the equation of the locus of \(P\).
(b) Give a geometric interpretation of the locus of \(P\).
WJEC Further Unit 1 2023 June Q6
6 marks Standard +0.3
6. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram. Given that $$| z - 3 + \mathrm { i } | = 2 | z - 5 - 2 \mathrm { i } |$$ show that the locus of \(P\) is a circle and write down the coordinates of its centre.
WJEC Further Unit 1 Specimen Q5
9 marks Standard +0.3
5. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram and $$| z - 3 | = 2 | z + \mathrm { i } |$$ Show that the locus of \(P\) is a circle and determine its radius and the coordinates of its centre.
Edexcel FP2 AS Specimen Q3
8 marks Standard +0.3
  1. A curve \(C\) is described by the equation
$$| z - 9 + 12 i | = 2 | z |$$
  1. Show that \(C\) is a circle, and find its centre and radius.
  2. Sketch \(C\) on an Argand diagram. Given that \(w\) lies on \(C\),
  3. find the largest value of \(a\) and the smallest value of \(b\) that must satisfy $$a \leqslant \operatorname { Re } ( w ) \leqslant b$$
Edexcel FP2 2019 June Q1
5 marks Standard +0.3
  1. A complex number \(z = x + \mathrm { i } y\) is represented by the point \(P\) in an Argand diagram.
Given that $$| z - 3 | = 4 | z + 1 |$$
  1. show that the locus of \(P\) has equation $$15 x ^ { 2 } + 15 y ^ { 2 } + 38 x + 7 = 0$$
  2. Hence find the maximum value of \(| z |\)
Edexcel FP2 2022 June Q5
6 marks Standard +0.8
  1. The locus of points \(z\) satisfies
$$| z + a \mathrm { i } | = 3 | z - a |$$ where \(a\) is an integer.
The locus is a circle with its centre in the third quadrant and radius \(\frac { 3 } { 2 } \sqrt { 2 }\) Determine
  1. the value of \(a\),
  2. the coordinates of the centre of the circle. \footnotetext{Question 5 continued }
Edexcel FP2 Specimen Q6
9 marks Challenging +1.2
  1. A curve has equation
$$| z + 6 | = 2 | z - 6 | \quad z \in \mathbb { C }$$
  1. Show that the curve is a circle with equation \(x ^ { 2 } + y ^ { 2 } - 20 x + 36 = 0\)
  2. Sketch the curve on an Argand diagram. The line \(l\) has equation \(a z ^ { * } + a ^ { * } z = 0\), where \(a \in \mathbb { C }\) and \(z \in \mathbb { C }\) Given that the line \(l\) is a tangent to the curve and that \(\arg a = \theta\)
  3. find the possible values of \(\tan \theta\)
Edexcel FP2 Q9
7 marks Standard +0.3
9. (a) The point \(P\) represents a complex number \(z\) in an Argand diagram. Given that $$| z - 2 \mathrm { i } | = 2 | z + \mathrm { i } |$$
  1. find a cartesian equation for the locus of \(P\), simplifying your answer.
  2. sketch the locus of \(P\).
    (b) A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is a translation \(- 7 + 11 \mathrm { i }\) followed by an enlargement with centre the origin and scale factor 3 . Write down the transformation \(T\) in the form $$w = a z + b , \quad a , b \in \mathbb { C }$$ [P6 June 2002 Qn 3]
AQA FP2 2015 June Q5
9 marks Standard +0.3
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)}
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