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2 Transformation Questions: Further Complex NumbersI need help with a couple of questions which can be found in Edexcel's AS and A Level Modular Mathematics FP2. Here it goes... 16. A transformation from the z-plane to the w-plane is defined by: $\displaystyle w=\frac { az+b }{ z+c } $, where a, b and c are elements of real numbers. Given that w=1 when z=0 and that w=3-2i when z=2+3i, a) Find the values of a, b and c, b) Find the exact values of the two points in the complex plane which remain invariant under the transformation. *I'm having particular problems with question b. |

As w = 1 when z = 0, b = c. As 3 - 2i = (a(2 + 3i) + b)/(b + 2 + 3i), 3b + 12 - (2b - 5)i = 2a + b + 3ai. Hence 3b + 12 = 2a + b and 2b - 5 = -3a. Solving those equations gives a = 17/5 and b = -13/5. The invariant points are the roots of the equation z = (az + b)/(z + c), which is equivalent to a quadratic equation. Solving it gives z = 3 ± 4√(2/5). |

For the second question, why does w suddenly become z? What's the purpose of this? May sound like a silly question - but I'd like to know. |

As the transformation is from the z-plane to the w-plane, the phrase "invariant under the transformation" means that w = z. |

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