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    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 + iy, to the w-plane, where w = u + iv, is given by

    w = \dfrac{4(1-i)z-8i}{2(-1+i)z-i} , \ 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 = \dfrac{ax^{2}+bxi+c}{16x^{2}+1}

    where a, b and c are real constants to be found.
    Spoiler:
    Show
    As given, y=x
    Therefore we can let z=x+ix
    Sub into w, simplify, then multiply by the complex conjugate of the denominator. (-4x+i)

    My answer: w = \dfrac{-32x^{2}+40xi+8}{16x^{2}+1}

    b) Hence show that the circle C has equation

    (u-3)^{2}+v^{2} = k^{2}

    where k is a constant to be found.


    ---------------------------------------------------

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

    Much appreciated.
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    (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:
    Show

    5
    or -5
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    (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:
    Show

    5
    or -5
    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
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    (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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