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Math Help - Problem solving coordinate geometry question

  1. #1
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    Problem solving coordinate geometry question

    Hello All,

    I wonder if anyone can help me with the question below.

    A line is drawn through the point A(1,2) to cut the line 2y = 3x - 5 in P and the line x + y = 12 in Q.

    If AQ = 2AP, find the coordinates P and Q.

    Try as I might, and I even have the answer to the value of P and Q, but I cant understand how anyone would be able to find out four unknowns (ie the cordinates of P and Q) with just those three equations.

    Thanks in advance!
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  2. #2
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    Re: Problem solving coordinate geometry question

    Spoiler Alert:

    The answer to P and Q for anyone who might be able to help if given the oppurtunuity to work backwards is...
    (\frac{2}{5}, \frac{-19}{10})
    and
    (2\frac{1}{5}, 9\frac{4}{5})
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  3. #3
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    Re: Problem solving coordinate geometry question

    Quote Originally Posted by Ahmadtaz View Post
    A line is drawn through the point A(1,2) to cut the line 2y = 3x - 5 in P and the line x + y = 12 in Q.
    If AQ = 2AP, find the coordinates P and Q.
    I would not like to solve this system. BUT.

    If P=(p_x,p_y)~\&~Q=(q_x,q_y) then

    2p_y=3p_x-5~\&~q_x+q_y=12 from incidence.

    4[(p_x-1)^2+(p_y-2)^2]=(q_x-1)^2+(q_y-2)^2 from distance.

    \frac{p_y-2}{p_x-1}= \frac{q_y-2}{q_x-1} from slope.
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  4. #4
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    Re: Problem solving coordinate geometry question

    Quote Originally Posted by Plato View Post
    I would not like to solve this system. BUT.

    If P=(p_x,p_y)~\&~Q=(q_x,q_y) then

    2p_y=3p_x-5~\&~q_x+q_y=12 from incidence.

    4[(p_x-1)^2+(p_y-2)^2]=(q_x-1)^2+(q_y-2)^2 from distance.

    \frac{p_y-2}{p_x-1}= \frac{q_y-2}{q_x-1} from slope.
    Sorry Plato, I see that you've put P and Y into the two equations for the lines, but could you explain how you get to this please?
    4[(p_x-1)^2+(p_y-2)^2]=(q_x-1)^2+(q_y-2)^2
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  5. #5
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    Re: Problem solving coordinate geometry question

    Apologies, I understand 4[(p_x-1)^2+(p_y-2)^2]=(q_x-1)^2+(q_y-2)^2 however, I don't understand how knowing the slope (your final equation) will help me solve the problem. Can you elaborate please?
    Last edited by Ahmadtaz; January 9th 2012 at 12:43 PM.
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  6. #6
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    Re: Problem solving coordinate geometry question

    Quote Originally Posted by Ahmadtaz View Post
    Apologies, I understand 4[(p_x-1)^2+(p_y-2)^2]=(q_x-1)^2+(q_y-2)^2 however, I don't understand how knowing the slope (your final equation) will help me solve the problem. Can you elaborate please?
    That gives you four equations in four unknowns.
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  7. #7
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    Re: Problem solving coordinate geometry question

    Quote Originally Posted by Ahmadtaz View Post
    Spoiler Alert:

    The answer to P and Q for anyone who might be able to help if given the oppurtunuity to work backwards is...
    (\frac{2}{5}, \frac{-19}{10})
    and
    (2\frac{1}{5}, 9\frac{4}{5})
    The answer should be P(4, 3.5) and Q(7,5). And it can be done in a easier method.

    Since AQ = 2AP, P is the mid-point of AQ.

    Let point Q=(x, 12-x)

    P, which is the mid-point, = (\frac{x+1}{2} , \frac{12-x+2}{2})

    Substitute P into the line 2y=3x-5, and you have

    2(\frac{14-x}{2}) = 3(\frac{x+1}{2})  - 5

    solve and you will get the points.
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  8. #8
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    Re: Problem solving coordinate geometry question

    Quote Originally Posted by earboth View Post
    I've done this question in a completely different way:

    1. Draw a sketch of the lines a:y = \frac32 x -5 and b:y=-x+12 and the point A.

    2. Determine the point of intersection of a and b: a\cap b = \{S\}

    3. Determine the equation of a line parallel to a through A p:y=\frac32 x +\frac12

    4. Determine the point of intersection of p and b: p\cap b = \{R\}

    5. You reach S if you go from R 2.2 units right and 2.2 units down. Now go from S 4.4 units right and 4.4 units down and you are at Q: Q\left(\frac{56}5\ ,\ \frac45 \right).

    6. Determine the equation of the line AQ = l: y = -\frac2{17} x + \frac{36}{17}. Determine the point of intersection of l \cap a=\{P\}

    7. I've got P\left(\frac{22}5\ ,\ \frac85 \right)
    Check your first equation. Should be.  y = \frac{3}{2} x - \frac{5}{2}
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