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Math Help - difficult complex numbers

  1. #1
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    difficult complex numbers

    The complex numbers Z and W are represented, respectively, by point P(x, y) and Q(u, v) in Argand diagrams and


    W = (Z+3)/(Z+1)

    The point P moves around the circle with equation |Z| = 1. Find the Cartesian equation of the locus of Q. Identify the locus.




    Any ideas??

    Thanks
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  2. #2
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    Quote Originally Posted by djmccabie View Post
    The complex numbers Z and W are represented, respectively, by point P(x, y) and Q(u, v) in Argand diagrams and


    W = (Z+3)/(Z+1)

    The point P moves around the circle with equation |Z| = 1. Find the Cartesian equation of the locus of Q. Identify the locus.




    Any ideas??

    Thanks
    First write the equation

    W=\frac{Z+3}{Z+1}

    in terms of W=u+iv and Z=x+iy and simplify and equate real and imaginary parts on the two sides of the equality sign.

    Now use the condition |Z|=1 to deduce x^2+y^2=1 and use this to eliminate x or y from the equations.

    Now simplify abd hopefully the equations you are left with for u and v in terms of x say will make it obvious what the locus is. To get the cartesian equation eliminate x between the two equations.

    CB
    Last edited by CaptainBlack; November 25th 2008 at 10:53 PM.
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  3. #3
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    Hi thanks for the reply, seems to be an easier method than my tutor provided except i dont understand a few things:

    W=\frac{Z+3}{Z+1}

    what is \frac? does that just mean write it as the fraction?

    Also:
    Now use the condition |Z|=1 to deduce x^2+y^2=1

    Does |Z| = x^2 + Y^2 ?

    I thought |Z| = root(x^2 + Y^2)


    Is there any chance you could show a solution? the work has been handed in and my tutor showed me his method but i dont fully understand.

    thanks
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  4. #4
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    Quote Originally Posted by djmccabie View Post
    Hi thanks for the reply, seems to be an easier method than my tutor provided except i dont understand a few things:

    W=\frac{Z+3}{Z+1}

    what is \frac? does that just mean write it as the fraction?

    Also:
    Now use the condition |Z|=1 to deduce x^2+y^2=1

    Does |Z| = x^2 + Y^2 ?

    I thought |Z| = root(x^2 + Y^2)


    Is there any chance you could show a solution? the work has been handed in and my tutor showed me his method but i dont fully understand.

    thanks
    The frac part is just a typo... it's LaTeX code to write fractions.

    If |Z| = \sqrt{x^2 + y^2} = 1

    Then squaring everything gives

    |Z|^2 = x^2 + y^2 = 1.

    So what Captain Black wrote is correct.
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    Quote Originally Posted by djmccabie View Post
    Hi thanks for the reply, seems to be an easier method than my tutor provided except i dont understand a few things:

    W=\frac{Z+3}{Z+1}

    what is \frac? does that just mean write it as the fraction?
    Sorry forgot to convert code to rendered mathematical text, fixed now.


    Also:
    Now use the condition |Z|=1 to deduce x^2+y^2=1

    Does |Z| = x^2 + Y^2 ?

    I thought |Z| = root(x^2 + Y^2)
    |Z|=1 if and only if |Z|^2=1, so |Z|=1 if and only if x^2+y^2=1

    CB
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  6. #6
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    You ask for an implementation, well we can start with:

    Quote Originally Posted by CaptainBlack View Post
    First write the equation

    W=\frac{Z+3}{Z+1}

    in terms of W=u+iv and Z=x+iy and simplify and equate real and imaginary parts on the two sides of the equality sign.
    W=u+iv=\frac{(x+3)+iy}{(x+1)+iy}

    ................ = \frac{[(x+3)+iy][(x+1)-iy]}{(x+1)^2+y^2}

    ................ = \frac{ [(x+3)(x+1)+y^2] +i[y(x+1)-y(x+3)]}{(x+1)^2+y^2}

    So equating real and complex parts:

     <br />
u=\frac{(x+3)(x+1)+y^2}{(x+1)^2+y^2}<br />

     <br />
v=\frac{-2y}{(x+1)^2+y^2}<br />

    CB
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    Quote Originally Posted by CaptainBlack View Post

    Now use the condition |Z|=1 to deduce x^2+y^2=1 and use this to eliminate x or y from the equations.
    Quote Originally Posted by CaptainBlack View Post
    So equating real and complex parts:

     <br />
u=\frac{(x+3)(x+1)+y^2}{(x+1)^2+y^2}<br />

     <br />
v=\frac{-2y}{(x+1)^2+y^2}<br />
    Now as x^2+y^2=1 we can write:

     <br />
u=\frac{(x^2+4x+3+y^2}{(x^2+2x+1+y^2}=\frac{4(x+1)  }{2(x+1)}=2<br />

     <br />
v=\frac{-2y}{2(x+1)}=\frac{-y}{x+1}<br />

    To simplify further observe that x^2+y^2=1 defines the unit circle so we may write:

    x=\cos(\theta),\ y=\sin(\theta)

    and simplify using this, or just observe that u is a constant and v can take any real value, so the locus is the line u=2.

    CB
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  8. #8
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    Using your method how would you solve this one?

    The complex numbers Z and W are represented, respectively, by points P(x,y) and Q(u, v) in argand diagrams and

    W=Z^2

    The point P moves along the line Y=x-1. Find the Cartesian equation of the locus of Q

    thanks
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