1. ## Transformation

A conic which gived by x^2+xy+y^2=3 Rotated 45 degrees about the origin counterclockwise. What is the quation is transformed to?

2. ## Re: Transformation

If we rotate the $x$ and $y$ axes (along with our conic) to new axes $x'$ and $y'$, then the old coordinates, in terms of the new ones, are:

$x = x'\cos 45^{\circ} + y'\sin 45^{\circ} = \dfrac{\sqrt{2}}{2}(x' + y')$

$y = y'\cos 45^{\circ} - x'\sin 45^{\circ} = \dfrac{\sqrt{2}}{2}(y' - x')$

So the equation in our new coordinates is:

$\left( \dfrac{\sqrt{2}}{2}(x' + y')\right)^2 + \left(\dfrac{\sqrt{2}}{2}(x' + y')\right)\left(\dfrac{\sqrt{2}}{2}(y' - x')\right) + \left(\dfrac{\sqrt{2}}{2}(y' - x')\right)^2 = 3$ or:

$\dfrac{1}{2}(x'^2 + 2x'y' + y'^2) + \dfrac{1}{2}(y'^2 - x'^2) + \dfrac{1}{2}(y'^2 - 2x'y' + x'^2) = 3$, and cancelling and collecting terms:

$x'^2 + 3y'^2 = 6$

Compare these two plots:

x^2+xy+y^2=3 - Wolfram|Alpha

x^2 + 3y^2 = 6 - Wolfram|Alpha

3. ## Re: Transformation

The answe in the book is 3x^2+y^2=6

I saw the graph in wolframalpha that is your answer is horisontal transformation, and the boks answer is vertikal one????

4. ## Re: Transformation

Your book is right, I mis-read the question and rotated it CLOCKWISE.

If we rotate it counter-clockwise, the formula for $x',y'$ are:

$x = x'\cos 45^{\circ} - y'\sin 45^{\circ} = \dfrac{\sqrt{2}}{2}(x' - y')$

$y = y'\cos 45^{\circ} + x'\sin 45^{\circ} = \dfrac{\sqrt{2}}{2}(x' + y')$

$\left( \dfrac{\sqrt{2}}{2}(x' - y')\right)^2 + \left(\dfrac{\sqrt{2}}{2}(x' = y')\right)\left(\dfrac{\sqrt{2}}{2}(x' + y')\right) + \left(\dfrac{\sqrt{2}}{2}(x' + y')\right)^2 = 3$

$\dfrac{1}{2}(x'^2 - 2x'y' + y'^2) + \dfrac{1}{2}(x'^2 - y'^2) + \dfrac{1}{2}(x'^2 + 2x'y' + y'^2) = 3$

$3x'^2 + y'^2 = 6$, as your book says, with this plot:

3x^2 + y^2 = 6 - Wolfram|Alpha

I apologize for the mix-up, I "turned it the wrong way"

5. ## Re: Transformation

Hi,
Deveno's answer is correct. Here's the way I like to think about such problems:

6. ## Re: Transformation

Thanks for explaining Transformation equation so properly.
It will definitely going to help folks who are getting an issue related transformation.