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Math Help - More geometry

  1. #1
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    More geometry

    A and B are points of intersection of y=x^2-2x and y=2-3x
    simultaneous equation solved = x^2 + x -2 = 0

    a) Find the coordinates of A and B, given that the x coordinate of B is greater than that of A.

    The line L is perpendicular to y=2-3x and passes through B.
    b) Find an equation of L

    The line M is parallel to y=5-3x and passes through A
    c) Find an equation of M

    The lime M crosses the y-axis at point P. The line L crosses the x-axis at point Q.
    d) Find the distance of PQ, giving your answer in square root form where a and b are integers and b is prime. (Hint - use pythagoras' theorem)
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by x-disturbed-x
    A and B are points of intersection of y=x^2-2x and y=2-3x
    simultaneous equation solved = x^2 + x -2 = 0

    a) Find the coordinates of A and B, given that the x coordinate of B is greater than that of A.
    You have identified correctly that the x co-ordinatea of the
    points of intersection of A and B are the solutions of

    x^2+x-2\ =\ 0

    so now find those solutions, let's call them x_1 and x_2, and we
    will assume that x_1\ <\ x_2, (they are not equal as the LHS
    of the equation is not the square of a single linear factor).

    Then from what we are told we know that the
    x co-ordinate of A is x_1, and the x co-ordinate of B
    is x_2.

    To find the y co-ordinates of A and B respectivly we now substitute
    x_1 and x_2 into

    y\ =\ 2-3x

    RonL
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by x-disturbed-x
    The line L is perpendicular to y=2-3x and passes through B.
    b) Find an equation of L
    The gradient of any line perpendicular to the line:

    y\ =\ m.x\ +\ c

    is -1/m.

    So you have sufficient information to determine the slope
    of L. Also you now know the co-ordinates of B (as you
    will have found them in part a), you can find in intercept
    for L by plugging the co-ordinates of B into into the
    general equation of a line with the required slope.

    RonL
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  4. #4
    Grand Panjandrum
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    [QUOTE=x-disturbed-x]The line M is parallel to y=5-3x and passes through A
    c) Find an equation of M

    This is like part b, except that the slope of M can be
    found using the result that the slopes of parallel lines are equal.

    Then proceed as in part b.

    RonL
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by x-disturbed-x

    The lime M crosses the y-axis at point P. The line L crosses the x-axis at point Q.
    d) Find the distance of PQ, giving your answer in square root form where a and b are integers and b is prime. (Hint - use pythagoras' theorem)
    You should by now have the equations for M and L, let these be:

    \mbox{L: }\ \ y\ =\ m_L.x\ +\ c_L

    \mbox{M: }\ y\ =\ m_M.x\ +\ c_M

    M crosses the y-axis when x\ =\ 0, which is at

    y\ =\ c_M

    so P is the point (0,c_M).

    Similarly L crosses the x-axis when y\ =\ 0, which is at

    x\ =\ -c_L/m_M

    so Q is the point (-c_L/m_L,0).

    So now you should know where P and Q are
    and so you can find the distance between them in the usual manner.

    RonL
    Last edited by CaptainBlack; December 1st 2005 at 09:16 AM.
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