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    Hello,

    Please solve the following...

    u = (6-3i)/(1+2i)

    i. For complex number z satisfying arg(z-u) = pi/4. , find the least value of |z|.

    ii. For complex numbers satisfying |z-(1+u)| = 1, find the greatest possible value of |z|.

    Thanks. ...
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    (Original post by methewthomson)
    Hello,

    Please solve the following...

    u = (6-3i)/(1+2i)

    i. For complex number z satisfying arg(z-u) = pi/4. , find the least value of |z|.

    ii. For complex numbers satisfying |z-(1+u)| = 1, find the greatest possible value of |z|.

    Thanks. ...
    First write u in the form

     u = a+bi

    It should then be clear how to draw the sketches
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    (Original post by Indeterminate)
    First write u in the form

     u = a+bi

    It should then be clear how to draw the sketches

    Thank you for the reply.

    Yes, I had some idea that first it should be written in a+bi form. But what is next? That is what I don't get....

    Could you please solve it out....perhaps with graphical illustration...
    thanks again.
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    (Original post by methewthomson)
    Thank you for the reply.

    Yes, I had some idea that first it should be written in a+bi form. But what is next? That is what I don't get....

    Could you please solve it out....perhaps with graphical illustration...
    thanks again.
    Well, take

    z=x+iy

    and draw the half-line representing arg(z-u)

    Deduce that \tan(\frac{\pi}{4}) =...... = 1

    As a further hint, you will need to find the minimum value of the quadratic

    |z|^2 = ....

    and then take the square root of that.
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    (Original post by Indeterminate)
    Well, take

    z=x+iy

    and draw the half-line representing arg(z-u)

    Deduce that \tan(\frac{\pi}{4}) =...... = 1

    As a further hint, you will need to find the minimum value of the quadratic

    |z|^2 = ....

    and then take the square root of that.
    Thank you again for replying.
    Actually, I'm bad at loci drawings and that is why I could not draw the half line representing arg(z-u). And why do we draw a half line?Could you plzzzz explain?

    u = 3i in our case. That is okay. But the next steps I don't understand.
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    (Original post by methewthomson)
    Thank you again for replying.
    Actually, I'm bad at loci drawings and that is why I could not draw the half line representing arg(z-u). And why do we draw a half line?Could you plzzzz explain?

    u = 3i in our case. That is okay. But the next steps I don't understand.
    u = -3i

    Can you draw easier loci?

    arg(z)=\frac{\pi}{3} for example.
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    You would get b+3=a. (1)
    if u assume a+ib as z
    now you have to minimise a^2 +b^2 =f(x)^2
    so put a from 1 and then differentiate to get b then find ur answer.
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    (Original post by Gifted)
    You would get b+3=a. (1)
    if u assume a+ib as z
    now you have to minimise a^2 +b^2 =f(x)^2
    so put a from 1 and then differentiate to get b then find ur answer.
    I can't say I understand what you're trying to do there.

    The question certainly doesn't require any differentiation.

    The thread title refers to loci. Once the correct loci are sketched a little bit of high school maths is all that is needed.
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    (Original post by BabyMaths)
    u = -3i

    Can you draw easier loci?

    arg(z)=\frac{\pi}{3} for example.

    Yes, I know how to draw that, but just that. I don't know how to draw the complex type.

    Can you plz explain the question I put on this thread?
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    (Original post by methewthomson)
    Yes, I know how to draw that, but just that. I don't know how to draw the complex type.

    Can you plz explain the question I put on this thread?
    You're trying to do a sketch for arg(z+3i)=\frac{\pi}{4}.

    Can you sketch arg(w)=\frac{\pi}{4}?

    Can you then use the relationship between z and w, namely z=w-3i?
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    (Original post by BabyMaths)
    You're trying to do a sketch for arg(z+3i)=\frac{\pi}{4}.

    Can you sketch arg(w)=\frac{\pi}{4}?

    Can you then use the relationship between z and w, namely z=w-3i?
    Thank you so very much for replying again.

    Yes I can sketch easily arg(w)=\frac{\pi}{4}. But how will I find the smallest value of w so that z turns out to be smallest, giving smallest modulus???

    Could you plz solve the question step by step?Your manner of explanation is really good.

    Plz help me as I lose lots of marks on such questions...
    thanks....
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    (Original post by methewthomson)
    Thank you so very much for replying again.

    Yes I can sketch easily arg(w)=\frac{\pi}{4}. But how will I find the smallest value of w so that z turns out to be smallest, giving smallest modulus???

    Could you plz solve the question step by step?Your manner of explanation is really good.

    Plz help me as I lose lots of marks on such questions...
    thanks....
    Post your sketch(es).

    I'd like to see them before I say any more.
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    (Original post by BabyMaths)
    Post your sketch(es).

    I'd like to see them before I say any more.
    Okay, here is the sketch you asked me to draw:
    Name:  untitled.jpg
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    (Original post by methewthomson)
    ...
    Try to re-upload your picture, doesn't seem to be working for me.
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    (Original post by Joshmeid)
    Try to re-upload your picture, doesn't seem to be working for me.
    Now you will be able to see a small picture. Open the picture in a new window to increase its size.
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    (Original post by methewthomson)
    Now you will be able to see a small picture. Open the picture in a new window to increase its size.
    Ok in that image, you've drawn \arg{z} = \frac{\pi}{4}. (ignoring the -3i and 3 marked on the axis.)

    This is the half-line starting from the origin with an angle of \frac{\pi}{4}.

    The argument is written in the form \arg{(z - a)} = \theta.

    In your case we have \arg{(z + 3i)} = \theta, which can be re-written as \arg{(z - -3i)} = \theta.

    We can then represent this on an argand diagram starting at the point (0, -3), with an angle of \theta from the positive x-axis.


    Now if you understand that, can you make another sketch of \arg{z - 4 + 2i} = \frac{3\pi}{4}.
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    (Original post by Joshmeid)
    Ok in that image, you've drawn \arg{z} = \frac{\pi}{4}.
    He's drawn arg(w)=pi/4 as I asked him.

    Now, if w=z+3i you have z=w-3i. Just sketch it in 3i below w.
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    (Original post by BabyMaths)
    He's drawn arg(w)=pi/4 as I asked him.

    Now, if w=z+3i you have z=w-3i. Just sketch it in 3i below w.
    I attempted to move away from that to prevent confusion later on when he arrives at transformations from z to w plane but feel free to continue.
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    (Original post by BabyMaths)
    He's drawn arg(w)=pi/4 as I asked him.

    Now, if w=z+3i you have z=w-3i. Just sketch it in 3i below w.

    Should I sketch z in 3i below? At what angle? Parallel to w?
    If possible please draw a diagram.
    Thanks.
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    (Original post by Joshmeid)
    I attempted to move away from that to prevent confusion later on when he arrives at transformations from z to w plane but feel free to continue.
    Thank you for all your efforts.
 
 
 
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