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Math Help - Help wih parametric representation of a function

  1. #1
    Newbie Rascot's Avatar
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    Post Help wih parametric representation of a function

    Hi, I'm stuck trying to obtain the cartesian equation of the function defined parametrically by x=1/2(t+1/t), y=1/2(t-1/t).

    The book (Mathematics for Engineers, Croft / Davison) gives x^2 - y^2 = 1 as the answer, but I can't figure out how they got there. Ideas?
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  2. #2
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    Re: Help wih parametric representation of a function

    Quote Originally Posted by Rascot View Post
    Hi, I'm stuck trying to obtain the cartesian equation of the function defined parametrically by x=1/2(t+1/t), y=1/2(t-1/t).

    The book (Mathematics for Engineers, Croft / Davison) gives x^2 - y^2 = 1 as the answer, but I can't figure out how they got there. Ideas?
    1. Rearrange both equations to:

    2x = t +\frac1t~\wedge~2y=t-\frac1t

    2. Add both sides of the equations:

    2x+2y = 2t~\implies~t = x+y

    3. Replace t in the 1st equation by (x+y):

    x = \frac12 \cdot \left((x+y)+\frac1{x+y} \right)

    4. Expand the brackets and after a few steps of rearranging you'll get the given result.
    Thanks from Rascot
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  3. #3
    Newbie Rascot's Avatar
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    Re: Help wih parametric representation of a function

    Perfect! Completely forgot about solving simultaneous equations (probably because they are dealt with later in the book). You have my thanks, sir.

    By the way, how do you (and pretty much everyone on this forum) post math symbols as images? Is there a piece of software you guys use?
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  4. #4
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    Re: Help wih parametric representation of a function

    Quote Originally Posted by Rascot View Post
    Perfect! Completely forgot about solving simultaneous equations (probably because they are dealt with later in the book). You have my thanks, sir.

    By the way, how do you (and pretty much everyone on this forum) post math symbols as images? Is there a piece of software you guys use?
    1. Go to LaTeX Help Forum. You'll find 2 tutorials where you can find how to use the implemented Latex-compiler.

    2. If you need the synatx of a Latex command you'll find it probably here: Helpisplaying a formula - Wikipedia, the free encyclopedia
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    Re: Help wih parametric representation of a function

    Hello, Rascot!

    Hi, I'm stuck trying to obtain the cartesian equation of the function
    . . defined parametrically by: . \begin{Bmatrix}x &=& \frac{1}{2}(t+\frac{1}{t}) \\ \\[-3mm] y&=& \frac{1}{2}(t-\frac{1}{t}) \end{Bmatrix}

    The book (Mathematics for Engineers, Croft/Davison) gives x^2 - y^2 \,=\, 1 as the answer.

    Square the equations: . \begin{Bmatrix}x^2 &=& \frac{t^2}{4} + \frac{1}{2} + \frac{1}{4t^2} & [1] \\ \\[-4mm] y^2 &=& \frac{t^2}{4} - \frac{1}{2} + \frac{1}{4t^2} & [2] \end{Bmatrix}

    Subtract [2] from [1]: . . x^2 - y^2 \;=\;1
    Thanks from Rascot
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  6. #6
    Newbie Rascot's Avatar
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    Re: Help wih parametric representation of a function

    Thank you both very much for your help. I can't mark the thread as solved, but would kindly ask moderators to do so if they happen to come across it.
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