Question

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

Answer 1

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 `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.