Results 1 to 5 of 5

Math Help - Equations

  1. #1
    Junior Member ihmth's Avatar
    Joined
    Jan 2008
    Posts
    44

    Equations

    Let S denote the set of points of intersection of the hyperbola xy = 2 and the graph of y = cube root of x^3 - 20. How many lines of slope 1 pass through at least one point of S?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by ihmth View Post
    Let S denote the set of points of intersection of the hyperbola xy = 2 and the graph of y = cube root of x^3 - 20. How many lines of slope 1 pass through at least one point of S?
    I'm not sure I understand the last bit "How many lines of slope 1 pass through at least one point of S?" .....

    You can draw a line of any desired gradient (including m = 1) through any given point (including points in S).

    So I would've thought the answer would be equal to the number of points in S .....

    Do you mean how many tangents to either xy = 2 or y = \sqrt[3]{x^3 - 20} ....? Or, perhaps how many lines passing through two of the points in S ....?

    Anyway, you certainly need to know how many points are in S ......

    Solve x \, \sqrt[3]{x^3 - 20} = 2 \Rightarrow x^3 (x^3 - 20) = 8 \Rightarrow x^6 - 20 x^3 - 8 = 0.

    Let x^3 = w, say:

    w^2 - 20 w - 8 = 0 \Rightarrow w = 10 \pm \sqrt{108} = 10 \pm 6 \sqrt{3}.

    Therefore x^3 = 10 \pm 6 \sqrt{3} and so there are two solutions for x. Therefore there are two points in S.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,829
    Thanks
    123
    Quote Originally Posted by ihmth View Post
    Let S denote the set of points of intersection of the hyperbola xy = 2 and the graph of y = cube root of x^3 - 20. How many lines of slope 1 pass through at least one point of S?
    First calculate the points of intersection:

    \frac2x = \sqrt[3]{x^3-20}. Cube both sides, multiply by x. You'll get an equation:

    x^6-20x^3-8 = 0 . Use the substitution t = x

    t^2-20t-8=0~\implies~t=10\pm 6\sqrt{3} which will give only 2 x-values. Plug in these x-values into the equation of the hyperbola:

    P_1\left(1+\sqrt{3} , \sqrt{3}-1\right), P_2\left(1-\sqrt{3} , -\sqrt{3}-1\right)

    Now use point-slope-formula to calculate the equations of the lines passing through P_1 or through P_2.

    You'll get in both cases: y = x-2

    To answer your question: There is only one line passing through the whole set of intersection points.
    Attached Thumbnails Attached Thumbnails Equations-ihmth_intersect.jpg  
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,707
    Thanks
    626
    Hello, ihmth

    Let S denote the set of points of intersection of the hyperbola xy = 2
    . . and the graph of y \:=\:\sqrt[3]{x^3 - 20}

    How many lines of slope 1 pass through at least one point of S ?
    We have: . \begin{array}{ccccccc}xy \; =\;2 & \quad\Rightarrow\quad & y \; = \;\frac{2}{x} & {\color{blue}[1]}\\<br />
y \; = \; \sqrt[3]{x^3-20} & \quad\Rightarrow\quad & x^3-y^3 \; = \; 20 & {\color{blue}[2]}\end{array}

    Substitute [1] into [2]: . x^3 - \left(\frac{2}{x}\right)^3 \:=\:20\quad\Rightarrow\quad x^6 - 20x^3 - 8 \;=\;0

    Quadratic Formula: . x^3 \;=\;\frac{20 \pm\sqrt{432}}{2} \;=\;10 \pm 6\sqrt{3} \;=\;(1\pm\sqrt{3})^3

    . . Hence: . x \:=\:1 \pm\sqrt{3}\quad\Rightarrow\quad y \:=\:-1\pm\sqrt{3}


    Therefore: . S has two points: . P\left(1\!+\!\sqrt{3},\:\text{-}1\!+\!\sqrt{3}\right),\;Q\left(1\!-\!\sqrt{3},\:\text{-}1\!-\!\sqrt{3}\right)


    Since the slope of PQ is: . m_{_{PQ}} \;=\;\frac{(\text{-}1-\sqrt{3}) - (\text{-}1+\sqrt{3})}{(1-\sqrt{3}) - (1+\sqrt{3})} \;=\; \frac{-2\sqrt{3}}{-2\sqrt{3}} \;=\;1

    . . there is one line with slope 1 that passes through P\text{ or }Q.

    Follow Math Help Forum on Facebook and Google+

  5. #5
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by mr fantastic View Post
    I'm not sure I understand the last bit "How many lines of slope 1 pass through at least one point of S?" .....

    You can draw a line of any desired gradient (including m = 1) through any given point (including points in S).

    So I would've thought the answer would be equal to the number of points in S .....

    Do you mean how many tangents to either xy = 2 or y = \sqrt[3]{x^3 - 20} ....? Or, perhaps how many lines passing through two of the points in S ....?

    [snip]
    I see nothing to change my opinion here ..... The question is open to several interpretations .....
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: April 7th 2010, 02:22 PM
  2. Equi Potential Equations or Euler Type Equations
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: April 2nd 2010, 02:58 AM
  3. Replies: 3
    Last Post: February 27th 2009, 07:05 PM
  4. Replies: 1
    Last Post: September 1st 2007, 06:35 AM
  5. Replies: 1
    Last Post: July 29th 2007, 02:37 PM

Search Tags


/mathhelpforum @mathhelpforum