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Math Help - Comple numbers and circles

  1. #1
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    Comple numbers and circles

    Let S be the interior of the circle |z 1 i| = 1. Show, by using suitable inequalities for |z1 ± z2|, that if z S then 5 1 < |z 3| < 5 + 1.


    Obtain the same result geometrically by considering the line containing the centre of the circle and the point 3.

    I'm not making headway with this question, so please help, thanks.
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by berachia View Post
    Let S be the interior of the circle |z 1 i| = 1. Show, by using suitable inequalities for |z1 ± z2|, that if z S then 5 1 < |z 3| < 5 + 1.
    I don't think that this is exactly right, it should be \sqqrt{5}-1\leq |z-3|\leq \sqrt{5}+1

    Obtain the same result geometrically by considering the line containing the centre of the circle and the point 3.

    I'm not making headway with this question, so please help, thanks.
    So for the algebraic part you start with |z-3| and transform it in a way that brings |z-(1+i)|=1 into play, like this:

    (*)\qquad |z-3|=|(z-(1+i))+(-2+i)|

    Now, you should remember that |z_1+z_2|\leq |z_1|+|z_2| (triangle inequality) and, similarly, ||z_1|-|z_2||\leq |z_1+z_2| (a consequence of the triangle inequality).

    Now set z_1 := z-(1+i) and z_2 := -2+i in (*) to get

    |z-3|\leq |z-(1+i)|+|-2+i|=1+\sqrt{5},

    because we are allowed to assume that |z-(1+i)|=1, and

    \sqrt{5}-1=||z-(1+i)|-|-2+i||\leq |z-3|

    For the geometrical part: the values of z that satisfy |z-(1+i)|=1 lie on a circle with center 1+i and radius 1, the values of z that satisfy |z-3|=\sqrt{5}\pm 1 lie on one of the two circles with center 3 and radii \sqrt{5}\pm 1.
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