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Points that are symmetric with respect to a circle C

  1. Nov 22, 2008 #1
    1. The problem statement, all variables and given/known data

    Lemma 1: Fix the circle C with center (x nought, y nought); y nought is greater than 0 and
    radius R is less than y nought. Consider two points P (x nought, y noight tilde) and P prime (x nought, -y nought tilde) which are symmetric with respect to x-axis by construcion.

    Prove that P and P prime are also symmetric with respect to Circle C if and only if y nought tilde is equal to the sqrt (y nought squared - radius squared).

    2. Relevant equations
    on the attachment



    3. The attempt at a solution
    I believe that these two points are orthogonal with respect to the x-axis. and they are symmetric because they fall on the same ray.
     

    Attached Files:

  2. jcsd
  3. Nov 23, 2008 #2

    tiny-tim

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    Welcome to PF!

    Hi brcole! Welcome to PF! :smile:

    You're making this too complicated …

    all the points are on the same "vertical" line …

    so just give every point a name, don't use || of vectors, just talk about the length of lines …

    Call (x0,0) Q.

    Then you have to prove that CP.CP´ = r2,

    and you know what CQ and QP and QP´are … :smile:
     
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