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Least distance between two complex numbers on two loci

  1. Apr 19, 2017 #1
    1. The problem statement, all variables and given/known data
    This is a CIE A'level maths P3 question out of an exam from 2013 in October/November. As there is no markscheme ( I at least can't find one), I would be grateful if someone could look at my solution to the problem and correct me if I made a mistake.
    The problem is 8.(b) below.
    IMG_1880.PNG
    2. Relevant equations


    3. The attempt at a solution
    The first locus they are asking for is that of a circle with centre (0,-1) and radius 1 and the second locus is a line 135 deg. to the horizontal (real number axis) starting at x=2. I call z1 and z2 the points which will give the least value of abs(z-w). Both these points must lie on a line l2. My further working and sketch of loci and the line are in the following image.In the last step I use the distance formula for the two complex numbers I calculated in the earlier steps. In the earlier steps I equated the equation of l2 and the equations for the loci.
    IMG_1908.JPG

    Thanks for any effort! And sorry for the clumsy exposition!
     
    Last edited: Apr 19, 2017
  2. jcsd
  3. Apr 19, 2017 #2

    haruspex

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    I get the same result. You can simplify it to √2 + 1/√2 - 1
     
  4. Apr 20, 2017 #3
    Thanks! That is great! Is my method correct and is there any general method how to approach these problems geometrically or is it just case by case observation? I know of a calculus based approach. As far as I know one expresses abs(z-w) in terms the general coordinates of any point on the loci and then sets the derivative equal to zero. Is this correct? And how would the general form for abs(z-w) look like? Thanks for any further effort?
     
  5. Apr 20, 2017 #4

    haruspex

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    A general method (not just circles and lines) will necessarily be by calculus. Of course, minimising |z-w| is the same as minimising |z-w|2, which simplifies things a little.
    In many cases, it will be a bit easier with a geometric approach. In this one, I did it by rotating the circle's centre through 45 degrees about the origin. Then I only needed the horizontal distance from the circle to the line x=√2.
     
  6. Apr 20, 2017 #5
    Thanks a lot ! And interesting approach. I will keep that in mind.
     
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