What does XY represent in an equation that contains the square of X and Y?

Question

For instance, an equation of an ellipse has equation of the form $\frac { x^2} {2^2}+\frac {y^2} {1^2}=1 $ is an ellipse centered at the origin with a horizontal major axis. However, I noticed that equations of ellipses that contain xy term don’t have horizontal major axises and they look like a rotated ellipse.

my question

• How does xy term affect the graph ? And does it have this affect to all other kind of graphs ? like hyperbola,circles,... And if there is a Geometrical interpretation for this Term.

Answer 1

$$f(x,y)=ax^2+by^2+cxy$$

Let apply the rotation $\begin{cases}x=\cos(t)u+\sin(t)v\\y=-\sin(t)u+\cos(t)v\end{cases}$

Then $f(u,v)=\underbrace{(\cdots)}_Au^2+\underbrace{(\cdots)}_Bv^2+\Big(\underbrace{c\,\cos(2t)-(a-b)\sin(2t)}_C\Big)uv$

If we take $t=\frac 12\tan^{-1}(\frac c{a-b})$ then $C=0$ and we get $$f(u,v)=Au^2+Bv^2$$

Which is an ellipse with axis parallel to coordinate axis in $(u,v)$ base.

So as long as the sign of $A,B$ is unchanged then the effect of the $xy$ term in the equation is some rotation combined with some distortion. Since it changes also the values of $A,B$ the ellipse get elongated either in one coordinate or the other.

But beware, it may happen that if $c$ is large enough compared to $a,b$ then one or both coefficients $A,B$ will change sign.

So instead of an ellipse $|A|u^2+|B|v^2=cst$ you might get an hyperbola $|A|u^2-|B|v^2=cst$ (try it on Desmos, https://www.desmos.com/calculator/orpaclx455 make $c$ vary).

Answer 2

By completing the square, you can group $xy$ with another term, for instance

$$x^2 + xy + y^2 = 5 \iff (x^2 + xy + \tfrac14 y^2) + \tfrac34 y^2 = 5 \iff (x + \tfrac12y)^2 + \tfrac34 y^2 = 5.$$

This means that:

• The transformation $(x,y) \mapsto (x + \frac12y, y)$ takes points satisfying $x^2+xy+y^2=5$ to points satisfying $x^2 + \frac34 y^2 = 5$: an axis-aligned ellipse we understand.
• The inverse transformation $(x,y) \mapsto (x - \frac12y, y)$ takes the axis-aligned ellipse $x^2 + \frac34 y^2 = 5$ to the ellipse $x^2 + xy + y^2 = 5$.

The transformation $(x,y) \mapsto (x - \frac12y, y)$ is a skew transformation that preserves points on the $x$-axis but shifts points above it to the left (and points below it to the right). It's a convenient way to draw the ellipse $x^2 + xy + y^2 = 5$ in Paint or similar software, which has an axis-aligned-ellipse tool and a skew tool (but no arbitrary ellipse tool).

Here's a diagram of the axis-aligned ellipse $x^2 + \frac34 y^2 = 5$ and the transformed ellipse $x^2 + xy + y^2 = 5$: