Equations with z/z* or zz* terms

Equations involving ratios like z/z*, products zz*, or fractions with z* in denominator, requiring use of |z|² = zz* or rationalization techniques before applying the standard substitution method.

3 questions

CAIE P3 2019 November Q7
7
  1. Find the complex number \(z\) satisfying the equation $$z + \frac { \mathrm { i } z } { z ^ { * } } - 2 = 0$$ where \(z ^ { * }\) denotes the complex conjugate of \(z\). Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    1. On a single Argand diagram sketch the loci given by the equations \(| z - 2 \mathrm { i } | = 2\) and \(\operatorname { Im } z = 3\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\).
    2. In the first quadrant the two loci intersect at the point \(P\). Find the exact argument of the complex number represented by \(P\).
AQA FP1 2011 June Q3
3 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$( z - \mathrm { i } ) \left( z ^ { * } - \mathrm { i } \right)$$
  2. Given that $$( z - \mathrm { i } ) \left( z ^ { * } - \mathrm { i } \right) = 24 - 8 \mathrm { i }$$ find the two possible values of \(z\).
Edexcel CP1 2022 June Q7
  1. Given that \(z = a + b \mathrm { i }\) is a complex number where \(a\) and \(b\) are real constants,
    1. show that \(z z ^ { * }\) is a real number.
    Given that
    • \(z z ^ { * } = 18\)
    • \(\frac { z } { z ^ { * } } = \frac { 7 } { 9 } + \frac { 4 \sqrt { 2 } } { 9 } \mathrm { i }\)
    • determine the possible complex numbers \(z\)