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Math Help - Locus

  1. #1
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    Locus

    I cant seem to do this questiono. Please give me a guide?

    A is a point where the circle with equation x^2+y^2=16 cuts the x-axis. Find the locus of the midpoints of all chords of this circle that contain the point A.
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  2. #2
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    Quote Originally Posted by Lukybear View Post
    I cant seem to do this questiono. Please give me a guide?

    A is a point where the circle with equation x^2+y^2=16 cuts the x-axis. Find the locus of the midpoints of all chords of this circle that contain the point A.
    1. Draw the circle (green line)
    2. Draw several (different) chords (blue lines) and their midpoints. Connect the midpoints by a smooth curve (see attachment)

    3. The point A is at A(4, 0) (Of course there is a second point A which will yield a different locus - but this part of the question is for you!)

    Point P(t, \sqrt{16-t^2}) is placed on the given circle.

    The midpoint of AP has the coordinates M\left(2-\frac12t\ ,\ \frac12\sqrt{16-t^2}\right)

    That means the locus of all midpoints is describend by the parametric equation:

    \left|\begin{array}{l}x=2-\frac12t \\ y= \frac12\sqrt{16-t^2}\end{array}\right.

    4. Determine t from the 1st equation: t = 2x-4
    Plug in this term into the 2nd equation:

    \begin{array}{rcl}y &=& \frac12\sqrt{16-(2x-4)^2} \\ y&=& \frac12 \sqrt{16x-4x^2} \\ 4y^2&=& 16x-4x^2 \\ (x-2)^2+y^2&=&2^2\end{array}
    Attached Thumbnails Attached Thumbnails Locus-krssehnenmitte.png  
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  3. #3
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    Wow thank you so much. However, is there a way without using parametric equations?
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