converting a conic equation to standard form.

Question

original equation: $1.1x^2-1.3xy+y^2-2.9x-1.9y = 0$

I found the angle of rotation to be 47.2 by finding the arccot of b/a-c

the x' and y' conversions:

$x = x'cos(47.2) - y'sin(47.2)$

$y = x'sin(47.2) + y'cos(47.2)$

the final equation once simplified is:

$.3981x^2+.0001xy-3.3643x + 1.7014y^2+.8368y = 0$

my question is how I would go about converting this into standard form so I can graph the elipsis on the rotated axis.

Answer 1

Since these are approximated values, forget the $.0001xy$ part. Then$$.3981x^2-3.3643x=(.630952x-2.66605)^2-7.10783$$and$$1.7014y^2+.8368y=(1.30438y+.320766)^2-.102891.$$

Answer 2

I would proceed as follows:

The discriminant $(-1.3)^2-4(1.1)(1.0)=-2.71\lt0$, so this is an ellipse. Its center can be found by differentiating and setting the partial derivatives to zero: $$\begin{align}2.2x-1.3y-2.9 &= 0 \\ -1.3x+2.0y-1.9 &= 0\end{align}$$ with solution $x=3.05166$, $y=2.93358$. So, first translate the origin to this point by making the substitution $x\mapsto x'+3.05166$, $y\mapsto y'+2.93358$, producing the transformed equation $$1.1 x'^2-1.3 x'y'+y'^2-7.21181=0.$$ You can save yourself some work by taking advantage of the fact that a translation doesn’t affect the second-degree terms and replaces the constant term with the value obtained by plugging the coordinates of the translation vector into the left-hand side of the equation; translating to the center of the ellipse eliminates the linear terms.

Now, you can apply your rotation to eliminate the cross term. This will leave the constant term unchanged, leaving the last step of normalizing the coefficients.