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Math Help - problem dealing with P.O.I. and equation of a line in 2d(impossible!!!)

  1. #1
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    problem dealing with P.O.I. and equation of a line in 2d(impossible!!!)

    hello, i recently had some homework to do for school and i came across a problem in the text that I couldn't solve so here is the problem and i will really appreciate it if somebody could solve it:

    the line segment joining A(2,3) to B(9,2) is the hyptenuse of a right triangle. The third vertex, C, lies on the line with these parametric equations:

    x = 2 + 2t
    y = 8 -t
    determine the coordinates of C.

    thanks a lot for any help!
    Last edited by dmitri; October 6th 2008 at 03:40 AM.
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  2. #2
    MHF Contributor
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    Determine the line where C is by eliminating the parameter t ....so that you will be dealing with x and y only, as the coordinates are in x and y only.

    y = 8 -t
    So, t = 8 -y
    Then,
    x = 2 +2t
    x = 2 +2(8 -y)
    x = 2 +16 -2y
    x = 18 -2y
    2y = -x +18
    y = (-1/2)x +9 ----------the line where C is.

    If AB is the hypotenuse of the right triangle with C as the right angle, then, by Pythagorean theorem,
    (AB)^2 = (AC)^2 +(BC)^2 ----------(i)

    Let C be point C(x,y).
    Umm, so it is simpler to express the x in terms of y.
    Since x = 18 -2y, so,
    C = C(18-2y,y)

    Then in (i),
    (9-2)^2 +(2-3)^2 = [((18-2y)-2)^2 +(y-3)^2] +[(18-2y)-9)^2 +(y-2)^2]
    Expand, simplify and solve for y.

    Then, x = 18 -2y.

    You should get two possible C's......(6,6) and (8,5).
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