Can someone help me convert these two?

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July 16th 2008, 07:12 PM
Dave53

Can someone help me convert these two?

Find the angle of rotation for the following conic that will eliminate the xy term. Rotate the axes through the angle of rotation, eliminate the xy term and write the conic in standard form.
xy=-10

Given the following point in polar form, change it to rectangular form.
(2,-(pi/3))

I'll be glad when this class is over!
July 16th 2008, 07:55 PM
Mathstud28

Quote:

Originally Posted by Dave53

Find the angle of rotation for the following conic that will eliminate the xy term. Rotate the axes through the angle of rotation, eliminate the xy term and write the conic in standard form.
xy=-10

Given the following point in polar form, change it to rectangular form.
(2,-(pi/3))

I'll be glad when this class is over!

For the second one we know that

$(x,y)\Rightarrow{(r\cos(\theta),r\sin(\theta)}$

so

$\left(2,-\frac{\pi}{3}\right)\Rightarrow\left(1,-\sqrt{3}\right)$
July 16th 2008, 07:59 PM
meymathis

Quote:

Originally Posted by Dave53

Find the angle of rotation for the following conic that will eliminate the xy term. Rotate the axes through the angle of rotation, eliminate the xy term and write the conic in standard form.
xy=-10

Given the following point in polar form, change it to rectangular form.
(2,-(pi/3))

I'll be glad when this class is over!

Well this has asymptotes perpendicular to the standard form so I would have guessed $\pi/4$ . Having not done this in years, if did some googeling (see http://www.mrjohns.net/ihs/0708/clas...Rotations3.pdf p. 4) I found that

$\phi = \frac{1}{2}cot^{-1}\left(\frac{A-C}{B}\right)$ Where A, B, C are the $x^2,\ xy,\ y^2$ terms respectively.

So we get $\phi = \frac{1}{2}cot^{-1}\left(0\right)=\frac{1}{2}\frac{\pi}{2}=\frac{\p i}{4}$

So then let
$x=x'\cos \phi-y' \sin \phi$
$y=x'\sin \phi+y' \cos \phi$

Howzat?