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    Hello, helpful math folk, I am stuck on question 14 p.43 of heinemann book, part b:

    Given that arg((z-1)/(z+1)) = pi/4

    (a) show that thte point P, which represents z on an Argand diagram, lies on an arc of a circle

    (b) Find the centre and radius of the circle and sketch the locus of P

    All help appreciated, as time running out, thanks!
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    (Original post by Rixius)
    Hello, helpful math folk, I am stuck on question 14 p.43 of heinemann book, part b:

    Given that arg((z-1)/(z+1)) = pi/4

    (a) show that thte point P, which represents z on an Argand diagram, lies on an arc of a circle

    (b) Find the centre and radius of the circle and sketch the locus of P

    All help appreciated, as time running out, thanks!
    Ther was a question similar to this yesterday.
    See here
    Look at the link in post #2. I think that may help.
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    It was the same area of p6, but i still dont know how to find the centre of the circle
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    (Original post by Rixius)
    It was the same area of p6, but i still dont know how to find the centre of the circle
    Well, if you draw the argand diagram, you can work that out from trig, i would think.
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    Ive just seen that im too late, but ive done it now.

    This is quite a long question, maybe ive used a stupid method, but anyways
    When you multiply compex numbers, you add the arguments, so multiply out the z+1 on the bottom;
    arg(z-1) = arg(z+1) + pi/4

    z = a + bi, arg(z) = arctan(b/a), so

    arctan(b/a-1) = arctan(b/a+1) + arctan(1)
    tan each side;
    b/(a-1) = [b/(a+1) + 1]/[1 - (b/a+1) *1]
    Simplify and multiply out to get

    a^2 + (b-1)^2 = 2
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    I did some trig and I got
    centre = (0,(1+√2)/3)
    radius = (2/3)√(3+1/√2)
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    I did it and got a different answer.

    If arg((z-1)/(z+1)) = pi/4

    then arg(z-1)-arg(z+1)=pi/4
    so the coords of A1,0) and B-1,0)
    The sector of the circle where the point P lies is therefore drawn from A to B where AP to BP is clockwise.

    Because the angle is pi/4,from circle laws we know the angle at the centre is twice the angle at the circumference, so wouldn't the angle at the centre be pi/2. If this is the case, I used trig to work out the opp (which would be the radius) and got r=2sin(pi/2)=2
    Using Pythagoras then to get the y coord of the height (because the x coord is 0): root of (1+2^2)

    So I get the centre to be (o,root5) and the radius to be 2.

    I don't have great credibility though, and would very much appreciate a genius to swoop in and tell me how to do this question too!
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    (Original post by JamesF)
    ...

    z = a + bi, arg(z) = arctan(b/a), so

    arctan(b/a-1) = arctan(b/a+1) + arctan(1)
    I didn't think you could do this with the arguments, but I've just done it the long way round and I get the same as you,

    x² + (y-1)² = 2

    So I must've messed up my trig!
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    (Original post by Hope)
    I did it and got a different answer.

    If arg((z-1)/(z+1)) = pi/4

    then arg(z-1)-arg(z+1)=pi/4
    so the coords of A1,0) and B-1,0)
    The sector of the circle where the point P lies is therefore drawn from A to B where AP to BP is clockwise.

    Because the angle is pi/4,from circle laws we know the angle at the centre is twice the angle at the circumference, so wouldn't the angle at the centre be pi/2. If this is the case, I used trig to work out the opp (which would be the radius) and got r=2sin(pi/2)=2
    Using Pythagoras then to get the y coord of the height (because the x coord is 0): root of (1+2^2)

    So I get the centre to be (o,root5) and the radius to be 2.

    I don't have great credibility though, and would very much appreciate a genius to swoop in and tell me how to do this question too!
    Check your trig - you should get the radius as √2 and the centre at y=1, x=0.
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    Oh yeh!! I used 2sin(pi/2) when it should have been 1/tan(pi/4)

    Then my answer follows to be centre at (0,1) radius root2.

    (heehee - my coordinates came out as smilies!! That'll teach me to use colons and brackets next to each other without wanting to make faces!)
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    (Original post by Hope)
    I don't have great credibility though, and would very much appreciate a genius to swoop in and tell me how to do this question too!
    How to do it!!

    arg[(z-z1)/(z-z2)] = pi/4

    where z1 = 1 and z2 = -1 and z = x + iy

    z-z1 = (x-1) + iy
    z-z2 = (x+1) + iy

    let z-z1 = r1ø1 and z-z2 = r2ø2

    ø1 = arg(z-z1) = arctan(y/(x-1))
    ø2 = arg(z-z2) = arctan(y/(x+1))

    then,

    (z-z1)/(z-z2) = (r1/r2)(ø1 - ø2)
    arg[(z-z1)/(z-z2)] = (ø1 - ø2) = pi/4

    take tan of both sides,

    (tanø1 - tanø2)/(1+tanø1tanø2) = tan(pi/4)
    (y/(x-1) - y/(x+1))/(1+y²/(x²-1)) = 1
    y(x+1) - y(x-1) = (x²-1) + y²
    yx + y - yx + y = x² - 1 + y²
    x² + y² - 2y - 1 = 0
    x² + (y-1)² - 2 = 0
    x² + (y-1)² = 2
    ==========

    centre at (0,1) r= √2
    =============

    Edit: of course, james F's method is far quicker!
 
 
 
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