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October 9th, 2018, 01:41 AM  #1 
Senior Member Joined: Jul 2013 From: United Kingdom Posts: 471 Thanks: 40  2 Transformation Questions: Further Complex Numbers
I 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 zplane to the wplane 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=32i 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. 
October 9th, 2018, 02:52 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,747 Thanks: 2133 
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). 
October 9th, 2018, 03:03 AM  #3 
Senior Member Joined: Jul 2013 From: United Kingdom Posts: 471 Thanks: 40 
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.

October 9th, 2018, 03:28 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,747 Thanks: 2133 
As the transformation is from the zplane to the wplane, the phrase "invariant under the transformation" means that w = z.


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