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Complex Numbers and Loci

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  1. Offline

    So I'm struggling with questions of the form:

    Find the locus of the point P, where P represents a complex number z such that \displaystyle \arg \left \frac{z - z_1}{z - z_2}\right = \alpha

    For instance:

    \displaystyle \arg \left \frac{z}{z + 4} \right = \frac{\pi}{4}

    As far as I can tell from answers to questions of this form, z describes the arc of a circle that starts at z_1 and goes clockwise round to z_2. Let's call these points A and B. The centre of the circle can be determined by noting that the angle AOB = 2\alpha.

    In the case of the example I gave (as far as I'm aware), the locus of the point P would look like this.

    My problem is that I don't have a clue why this is, or if this is even actually the case. If anybody could help me it would be very much appreciated.

  2. Offline

    Use arg(z/w)=argz-argw and then letting z=x+iy and considering the arg as arctan might help to see where the locus comes from.
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    i find it very difficult to see these loci without a diagram. hope this one helps


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