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 perfect_world October 9th, 2018 01:41 AM

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 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.

 skipjack October 9th, 2018 02:52 AM

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).

 perfect_world October 9th, 2018 03:03 AM

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.

 skipjack October 9th, 2018 03:28 AM

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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