How do I graph the ellipse with the equation $−x+2y+x^2+xy+y^2=0$ ?

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Answer 1 · 2015-07-09T15:50:15

Remove the cross-product term, then graph on the new coordinate axes.

There are various forms and formulas used for conic sections. I use $Ax^2 + Bxy +Cy^2 +Dx +Ey +F =0$

And $cot2theta = (A-C)/B$ or $tan2theta = B/(A-C)$

So we re-write:

$−x+2y+x^2+xy+y^2=0$ In the form:

$x^2 +xy +y^2 -x +2y =0$

We get: $cot2theta = (1-1)/1 = 0$ ,

so $2theta = pi/2 = 90^@$

and $theta = pi/4 = 45^@$

Our new coordinate system will be denoted $hat(x)$ and $hat(y)$ . To get this equations replace $x$ and $y$ by:

$x = hatxcostheta- haty sintheta$
$y = hatxsintheta+hatycostheta$

Using $theta = pi/4$ , we get:

$x = hatx/sqrt2- haty/sqrt2$

$y = hatx/sqrt2+haty/sqrt2$

Replace and simplify.

I get

$(3hatx)/2 + hatx/sqrt2 + haty^2/2 +(3haty)/sqrt2 = 0$

Now treat it like a non-rotated ellipse. Find center and vertices and graph on the new axes.

How do I graph the ellipse with the equation −x+2y+x^2+xy+y^2=0−x+2y+x^2+xy+y^2=0?