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    The point P resresents the complex number z, where
    |z+i|=sqrt(2)|z+(i/2)|

    The transformation T from the z-plane to w-plane is given by
    w=[z+isqrt(2)]/[izsqrt(2)+1]

    Express as a locus in the w-plane the image of P under the transformation T.

    I had shown in a previous part of the question that |z|=1/sqrt(2)

    |x+i(y+1)|=sqrt(2)|x+i(y+0.5)|
    x2+(y+1)2=2[x2+(y+0.5)2]
    x2+y2+2y+1=2x2+2y2+2y+0.5
    x2+y2=0.5
    sqrt(x2+y2)=|x+iy|=|z|=1/sqrt(2)
    ***
    w=[z+isqrt(2)]/[izsqrt(2)+1]

    w=[z+isqrt(2)]/[izsqrt(2)+1]
    wizsqrt(2)+w=z+isqrt(2)
    z[wisqrt(2)-1]=isqrt(2)-w
    z=[isqrt(2)-w]/[wisqrt(2)-1]

    1/sqrt(2)=|isqrt(2)-w|/|wisqrt(2)-1|
    |wisqrt(2)-1|=sqrt(2)|isqrt(2)-w|
    Not sure how to simplify this, and anyway, the answer page says |w-isqrt(2)|=|w+i/sqrt(2)|
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    (Original post by Nuclear Ghost)
    The point P resresents the complex number z, where
    |z+i|=sqrt(2)|z+(i/2)|

    The transformation T from the z-plane to w-plane is given by
    w=[z+isqrt(2)]/[izsqrt(2)+1]

    Express as a locus in the w-plane the image of P under the transformation T.

    I had shown in a previous part of the question that |z|=1/sqrt(2)

    |x+i(y+1)|=sqrt(2)|x+i(y+0.5)|
    x2+(y+1)2=2[x2+(y+0.5)2]
    x2+y2+2y+1=2x2+2y2+2y+0.5
    x2+y2=0.5
    sqrt(x2+y2)=|x+iy|=|z|=1/sqrt(2)
    ***
    w=[z+isqrt(2)]/[izsqrt(2)+1]

    w=[z+isqrt(2)]/[izsqrt(2)+1]
    wizsqrt(2)+w=z+isqrt(2)
    z[wisqrt(2)-1]=isqrt(2)-w
    z=[isqrt(2)-w]/[wisqrt(2)-1]

    1/sqrt(2)=|isqrt(2)-w|/|wisqrt(2)-1|
    |wisqrt(2)-1|=sqrt(2)|isqrt(2)-w|
    Not sure how to simplify this, and anyway, the answer page says |w-isqrt(2)|=|w+i/sqrt(2)|

    I find it hard to follow

    could you post a picture or tell us where the question comes from please.
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    Name:  1421161799702.jpg
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    Your solution is equivalent to the answer page, IMHO. The answer page has a non-rationalised denominator so might not be preferable.

    You can use the following rules to convert between them:
    |a|.|b| = |ab|
    |-z| = |z|

    Quick explanation here: from your answer, divide both sides by sqrt(2), and bring that into the left hand modulus. Multiply the left hand side by |-i| (allowed since |-i| = 1). Then 'negate' the right hand modulus - multiplying by |-1|.

    The advantage of the mark scheme answer is that you can now express the locus as a perpendicular bisector of two points, if I'm right. Good luck!
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    rearrange for z
    mod both sides
    LHS: |z|= 1/√2
    RHS: write w = u+iv
    rearrange to get Cartesian
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    (Original post by ronalddotgl)
    Your solution is equivalent to the answer page, IMHO. The answer page has a non-rationalised denominator so might not be preferable.

    You can use the following rules to convert between them:
    |a|.|b| = |ab|
    |-z| = |z|


    Quick explanation here: from your answer, divide both sides by sqrt(2), and bring that into the left hand modulus. Multiply the left hand side by |-i| (allowed since |-i| = 1). Then 'negate' the right hand modulus.

    The advantage of the mark scheme answer is that you can now express the locus as a perpendicular bisector of two points, if I'm right. Good luck!
    Seems to work. Thanks.
 
 
 
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