# Help wih parametric representation of a function

• Jun 5th 2012, 11:20 PM
Rascot
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?
• Jun 5th 2012, 11:44 PM
earboth
Re: Help wih parametric representation of a function
Quote:

Originally Posted by Rascot
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.
• Jun 6th 2012, 12:05 AM
Rascot
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?
• Jun 6th 2012, 12:26 AM
earboth
Re: Help wih parametric representation of a function
Quote:

Originally Posted by Rascot
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: Help:Displaying a formula - Wikipedia, the free encyclopedia
• Jun 6th 2012, 06:30 AM
Soroban
Re: Help wih parametric representation of a function
Hello, Rascot!

Quote:

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]: . . $x^2 - y^2 \;=\;1$
• Jun 6th 2012, 12:59 PM
Rascot
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.