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Math Help - Tricky Question - challenge

  1. #1
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    Cool Tricky Question - challenge

    my teacher said anyone in my class who gets this question would get a prize in front of the whole school.... okay.... i exaggerate.... but still, i want to understand this.

    the question is:

    consider the circle x^2 + y^2 = 25 and the line y = x +k, where k is any real number. Determine the values of k for which the line will intersect the circle in one, two, or no points. Repeat for the circle x^2 + y^2 = 49 and the line y = x+k.

    generalize your results for a circle of radius r and the line y = x+k

    Note: the question asks for you to do two. Dont worry about it, just explain the steps and i will understand it.

    Thanks in advance
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  2. #2
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    Quote Originally Posted by thejoester View Post
    consider the circle x^2 + y^2 = 25 and the line y = x +k, where k is any real number. Determine the values of k for which the line will intersect the circle in one, two, or no points. Repeat for the circle x^2 + y^2 = 49 and the line y = x+k.
    What class is this?

    Okay,
    If they intersect we can substitute y=x+k into the equation,
    x^2+(x+k)^2=25
    x^2+x^2+2xk+k^2=25
    2x^2+2xk+k^2=25
    2x^2+2xk+(k^2-25)=0
    To have 2 distinct points we need the discrimant to be positive.
    (2k)^2-4(2)(k^2-25)>0
    Thus,
    4k^2-8k^2+200>0
    200-4k^2>0
    50-k^2>0
    50>k^2
    k^2<50
    |k|<\sqrt{50}
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  3. #3
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    Hello, thejoester!

    Consider the circle x^2 + y^2 \,= \,25 and the line y \,= \,x +k
    Determine the values of k for which the line will intersect the circle in 1, 2, or 0 points.

    To complete what ThePerfectHacker started . . .

    The discriminant is: . D \:=\:4(50 - k^2)

    To have one point of intersection, D = 0
    . . 4(50 - k^2)\:=\:0\quad\Rightarrow\quad k^2 \,=\,50\quad\Rightarrow\quad k \,=\,\pm5\sqrt{2}

    To have two points of intersection: D > 0
    . . 4(50-k^2)\:>\:0\quad\Rightarrow\quad k^2 \:<\:50\quad\Rightarrow\quad |k| \:< \:5\sqrt{2}

    To have no points of intersection: D < 0
    . . 4(50-k^2) \:<\:0\quad\Rightarrow\quad k^2 \:>\:50\quad\Rightarrow\quad|k|\:>\:5\sqrt{2}


    The "graph" looks like this:

    . . \overbrace{- - - }^{\text{0 points}}+\overbrace{ - - - + - - - }^{\text{2 points}}+\overbrace{ - - -}^{\text{0 points}}
    . . . . . \text{-}5\sqrt{2} . . . . . . . . . 5\sqrt{2}
    . . . . . ^{\text{1 point}} . . . . . . . . . ^{\text{1 point}}

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  4. #4
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    Thanks!

    thanks a lot guys, i really appreciate it.
    i have one more question though, would either of you know how to graph this on excel?
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