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Math Help - parabola problems

  1. #1
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    parabola problems

    1- prove that the line X+Y=1 touches the parabola Y=X-X
    and what is the coordinates of tangency

    2-find the equation of the parabola having 3X+4Y-5=0 as directrix and (5,-6) as focus find it is vertex latux rectum and the equation of its axis and its tangent at the vertex
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  2. #2
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    Hello, mohamedsafy!

    1) Prove that the line x + y \:=\:1 touches the parabola y\:=\:x-x^2

    and find the coordinates of tangency.
    A line can intersect a parabola in 0, 1 or 2 points.


    We have: . \begin{array}{cccc}y &=& 1-x & {\color{blue}[1]}\\ y &=&x -x^2 & {\color{blue}[2]}\end{array}

    Equate [1] and [2]: . 1 - x \:=\:x-x^2\quad\Rightarrow\quad x^2-2x + 1 \:=\:0 \quad\Rightarrow\quad (x-1)^2 \:=\:0

    . . x-1\:=\:0 \quad\Rightarrow\quad x \:=\:1

    Substitute into [1]: . y \:=\:1- 1 \:=\:0

    The line intersects the parabola at (1,0).

    Since the line intersects the parabola in one point,
    . . the line is tangent to the parabola.

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  3. #3
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    thx

    thx for this i really apreciate it


    but couldnt anyone solve the second one ???????
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