I'm studying quadratic forms in algebra at the moment and I've been asked to plot the following curve: $$3x^2+4xy+3y^2-\sqrt{2}x+\sqrt{2}y=1$$
I have used the following transformations: $$x_1=x+\frac{2}{3}y$$ $$y_1=y$$ and then $$x_2=x_1-\frac{\sqrt{2}}{6}$$ $$y_2=y_1+\frac{\sqrt{2}}{2}$$
which leaves me with the following equation: $$3x_2^2+\frac{5}{3}y_2^2=2$$.
I can now see that I have an ellipse but I don't know how to work out what the centre of my original ellipse is as well as the major and minor radii. How would I do this?