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Further Complex Numbers - Transformations

Could someone walk me through part b? and check my answer/method to part a?


The transformation T from the z-plane, where z=x+iyz = x + iy, to the w-plane, where w=u+ivw = u + iv, is given by

w=4(1−i)z−8i2(−1+i)z−iw = \dfrac{4(1-i)z-8i}{2(-1+i)z-i} ,,  z≠14−14i\ z \not= \dfrac{1}{4} - \dfrac{1}{4}i

The transformation T maps the points on the line l with equation y=x in the z-plane to a circle C in the w-plane

a) Show that:

w=ax2+bxi+c16x2+1w = \dfrac{ax^{2}+bxi+c}{16x^{2}+1}

where a, b and c are real constants to be found.

Spoiler


b) Hence show that the circle C has equation

(u−3)2+v2=k2(u-3)^{2}+v^{2} = k^{2}

where k is a constant to be found.


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I don't know where to start.. yet (part b). I don't see it.

Much appreciated.
(edited 8 years ago)
Original post by edothero


I don't know where to start.. yet (part b). I don't see it.

Much appreciated.


Agree with part a).

Part b) w=u+iv

So separate your w into real and imaginary parts (for your u and v) and sub into LHS of given equation.

I get k=

Spoiler

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edothero
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15
Original post by ghostwalker
Agree with part a).

Part b) w=u+iv

So separate your w into real and imaginary parts (for your u and v) and sub into LHS of given equation.

I get k=

Spoiler



I see, the value of k checks out for me also.
Though can the answer realistically be -5? as we are dealing with the radius?

I appreciate your help
Original post by edothero
I see, the value of k checks out for me also.
Though can the answer realistically be -5? as we are dealing with the radius?


Fair point. 5 it is.

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