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Math Help - locus word problem

  1. #1
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    locus word problem

    A dog walking through a yard. Tree A(-2,3), Tree B(4, -6).

    Dog is at all times, twice as far from A as it from B.
    Draw a graph showing the trees, path of dog and relationship defining the locus.

    Write geometric description of the path of the dog, relative to the two trees.

    I had no idea where to start, so I plotted the points on the graph and did the equation for Midpoint of AB and plotted that point - but it looks SO OFF!

    \frac {-2+4}{2} , \frac{3+(-6)}{2}<br />
\\=1,4.5
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  2. #2
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    Quote Originally Posted by sux@math View Post
    A dog walking through a yard. Tree A(-2,3), Tree B(4, -6).

    Dog is at all times, twice as far from A as it from B.
    Draw a graph showing the trees, path of dog and relationship defining the locus.

    Write geometric description of the path of the dog, relative to the two trees.

    I had no idea where to start, so I plotted the points on the graph and did the equation for Midpoint of AB and plotted that point - but it looks SO OFF!

    \\=1,4.5" alt="\frac {-2+4}{2} , \frac{3+(-6)}{2}
    \\=1,4.5" />
    Let a general point on the locus of the dog be P(x, y). Then dPA = 2 dPB.

    So get the distance dPA between P and A and the distance dPB between P and B. Then substitute .....

    By the way, the locus is a Circle of Apollonius. Some sites that might interest:

    Circles of Apollonius

    The circle of Apollonius
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  3. #3
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    Hello, sux@math!

    A dog walking through a yard. Tree A(-2,3), Tree B (4, -6)

    Dog is at all times, twice as far from A as it is from B.

    Draw a graph showing the trees, path of dog and relationship defining the locus.

    Write geometric description of the path of the dog, relative to the two trees.
    Code:
          A   |             D
          * - + - - - - - - * (x,y)
       (-2,3) |            /
      --------+-----------/--------
              |          /
              |         /
              |        /
              |       * (4,-6)
              |       B

    The trees are at A\text{ and }B; the dog is at D(x,y)


    Since DA \:=\:2\!\cdot\!DB, we have: . \sqrt{(x+2)^2+(y-3)^2} \;=\;2\cdot\sqrt{(x-4)^2+(y+6)^2}

    Square both sides: . x^2 + 4x + 4 + y^2 - 6y + 9 \;\;=\;\;4(x^2-8x+16 + y^2 + 12y + 36)

    . . which simplifies to: . x^2 - 12x + y^2 + 18y \;=\;65


    Complete the square: . (x^2-12x \;{\color{blue}+\: 36}) + (y^2 + 18y\;{\color{red}+\: 81}) \;=\;65 \;{\color{blue}+\: 36}\:{\color{red} +\: 81}

    . . and we have: . (x-6)^2 + (y+9)^2 \;=\;182


    The dog's path is a circle with center (6,\:\text{-}9) and radius \sqrt{182}

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  4. #4
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    You really are a *super* member, Soroban! A million, trillion thanks!!!!
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  5. #5
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    ...further more, it is UNBELIEVABLY helpful to a visual learner like me that you put the different colours in your equation.

    Honestly, thanks so much for taking the time - you have no idea how much you've helped me with your efforts!
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  6. #6
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    Quote Originally Posted by sux@math View Post
    ...further more, it is UNBELIEVABLY helpful to a visual learner like me that you put the different colours in your equation.

    Honestly, thanks so much for taking the time - you have no idea how much you've helped me with your efforts!
    Did you try doing the problem using:
    1. The suggestions I gave?
    2. Taking a look at the links I provided and playing around with the dynamic geometry .....

    *Sigh* If only super members could be brought into exams like calculators can be. Alas .....
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  7. #7
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    Haha! Mr. Fantastic! I am so sorry if I offended you! To a math n00b like me, your answer seemed either a) glib or b) waaayyyyy over my head and the link gave me a small seizure, or c) both.

    Yes, I did try your suggestion but had no luck until I saw Soroban's post.

    I apologize from the bottom of my cold-black heart!
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