A equation of an ellipse can be a circle in another referential?

Question

I have 8*x^2-12*x*y+17*y^2=20. And i formed the squares and obtain: 8*x^2-12*x*y+17*y^2=20=(x-3y/4)^2+25*y^2/16=20/8. Now substitute x-3y/4=X and 5y/4=Y, put into the equation and obtain: X^2+Y^2=20/8 well this is a circle but ... 8*x^2-12*x*y+17*y^2=20. is an ellipse. That ellipse is a circle if i use other system of axes?

Answer 1

$$x^2+\frac{y^2}4=1$$ describes an ellipse of half-axis lengths $1$ and $2$.

Now let $X:=x,Y:=\dfrac y2$ and

$$X^2+Y^2=1$$ is the unit circle.

A non-orthogonal transform is anisotropic and stretches the shapes.