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

Jul 9, 2015

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

#### Explanation:

There are various forms and formulas used for conic sections. I use $A {x}^{2} + B x y + C {y}^{2} + D x + E y + F = 0$

And $\cot 2 \theta = \frac{A - C}{B}$ or $\tan 2 \theta = \frac{B}{A - C}$

So we re-write:

−x+2y+x^2+xy+y^2=0 In the form:

${x}^{2} + x y + {y}^{2} - x + 2 y = 0$

We get: $\cot 2 \theta = \frac{1 - 1}{1} = 0$,

so $2 \theta = \frac{\pi}{2} = {90}^{\circ}$

and $\theta = \frac{\pi}{4} = {45}^{\circ}$

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

$x = \hat{x} \cos \theta - \hat{y} \sin \theta$
$y = \hat{x} \sin \theta + \hat{y} \cos \theta$

Using $\theta = \frac{\pi}{4}$, we get:

$x = \frac{\hat{x}}{\sqrt{2}} - \frac{\hat{y}}{\sqrt{2}}$

$y = \frac{\hat{x}}{\sqrt{2}} + \frac{\hat{y}}{\sqrt{2}}$

Replace and simplify.

I get

$\frac{3 \hat{x}}{2} + \frac{\hat{x}}{\sqrt{2}} + {\hat{y}}^{2} / 2 + \frac{3 \hat{y}}{\sqrt{2}} = 0$

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