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Show that this locus determines a circle in C

  1. Feb 23, 2006 #1
    a and b represent two numbers on an Argand plane, i.e. fixed. Then it is given that locus of z given by the equation
    |(z - a)/(z - b)| = k where k is not 1.
    Now it is given that locos of z represents a circle. I cannot understand how can this equation represent a circle.
    |z -a| k|z - b| - this maks me think geometrically impossible to represent a circle.
    Can anyone help me here?
    The question is to find centre and radius.
  2. jcsd
  3. Feb 23, 2006 #2
    |z-a| k |z -b|
    it is not so
    |z - a| = k|z - b| - it was a typo.
  4. Feb 23, 2006 #3
    Geometrically, the equation is stating that the difference in angle between the two vectors is a constant, and so is their ratio of lengths. The former statement is more important. Consider a circle through the points a and b and the chord a-b. Consider z on the circle and the lines z-a and z-b. There is a simple theorem about the angle between those two lines for any point z on the circle. See http://mathworld.wolfram.com/Chord.html .
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