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Thread: plane geometry

  1. #1
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    plane geometry

    Question: A line is drawn through the point A (1,2) to the cut the line  2y=3x-5 in P and the line x + y = 12 in Q. If AQ = 2AP, find the coordinates of P and Q.

    I have drawn a sketch but desperately need help i cant figure out how to answer it
    plane geometry-bos-1.jpg
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  2. #2
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    Re: plane geometry

    For me, this was quite a problem. There's probably an easier way to solve this but the following sledge "hammer approach" works. I needed the computer to help me solve an equation.



    Here's a drawing of the situation with m=1/2:

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  3. #3
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    Re: plane geometry

    Let
    P=(a,b) and Q=(c,d)

    \overrightarrow{\text{AQ}}=\pm  2 \overrightarrow{\text{AP}}

    Translating into coordinates gives 2 linear equations

    \pm 2(a-1)=c-1

    \pm 2(b-2)=d-2

    Two more linear equations:

    2b=3a-5

    c+d=12

    EDIT: Looking at the sketch provided by the OP, APQ doesn't look at all like a straight line in which case the problem has infinitely many solutions. Otherwise, only two solutions
    Last edited by Idea; Mar 29th 2017 at 11:46 AM.
    Thanks from johng
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  4. #4
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    Re: plane geometry

    As usual, Idea gave a much nicer solution than mine. However, by hand his solution still requires some grunt work. Here are some observations:



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  5. #5
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    Re: plane geometry

    Quote Originally Posted by Idea View Post
    Let
    P=(a,b) and Q=(c,d)

    \overrightarrow{\text{AQ}}=\pm  2 \overrightarrow{\text{AP}}

    Translating into coordinates gives 2 linear equations

    \pm 2(a-1)=c-1

    \pm 2(b-2)=d-2

    Two more linear equations:

    2b=3a-5

    c+d=12

    EDIT: Looking at the sketch provided by the OP, APQ doesn't look at all like a straight line in which case the problem has infinitely many solutions. Otherwise, only two solutions
    where is the solution did you edit it
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  6. #6
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    Re: plane geometry

    Quote Originally Posted by johng View Post
    As usual, Idea gave a much nicer solution than mine. However, by hand his solution still requires some grunt work. Here are some observations:



    thank you very much for your help you went into detail for this problem. I truly appreciate your work.

    But can I ask how do you do the matrix as normal simultaneous equation as I have not yet being taught matrix
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  7. #7
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    Re: plane geometry

    Quote Originally Posted by johng View Post
    For me, this was quite a problem. There's probably an easier way to solve this but the following sledge "hammer approach" works. I needed the computer to help me solve an equation.



    Here's a drawing of the situation with m=1/2:

    I understand but I want to ask just one thing
    how do you get for line l: y=m(x-1) +2 where did it come from

    thank you
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