<s>[INST] What are the A,B,C parameters of this ellipse formula? I am looking at

$$A(x − h)^2 + B(x − h)(y − k) + C(y − k)^2 = 1$$

This is a rotating ellipse formula, where $h,k$ are the centroid of the ellipse. I have tried looking around for $A,B,C$ parameters, and I see that they are from Quadratic formula. But to be frank, I want to see how $A,B,C$ impact the orientation of the ellipse and I am looking for a more visual explanation for this question. 
 [/INST] I would like to explain the transforms on a picture.

$$\frac{x'^2}{a^2}+\frac{y'^2}{b^2}=1$$
$$x=x'.\cos \alpha - y'. \sin \alpha $$
$$y=x'.\sin \alpha + y'. \cos \alpha $$
We can get easily the result above from the picture 

$$\cos \alpha. x=\cos \alpha (x'.\cos \alpha - y'. \sin \alpha) $$
$$\sin \alpha. y=\sin \alpha (x'.\sin \alpha + y'. \cos \alpha) $$
$$x'=x.\cos \alpha + y. \sin \alpha $$

$$-\sin \alpha. x=\cos \alpha (x'.\cos \alpha - y'. \sin \alpha) $$
$$\cos \alpha. y=\sin \alpha (x'.\sin \alpha + y'. \cos \alpha) $$
$$y'=-x.\sin \alpha + y. \cos \alpha $$

$$\frac{(x.\cos \alpha + y. \sin \alpha)^2}{a^2}+\frac{(-x.\sin \alpha + y. \cos \alpha)^2}{b^2}=1$$
$$(\frac{\cos^2 \alpha }{a^2}+\frac{\sin^2 \alpha }{b^2})x^2+2(\frac{\cos \alpha \sin \alpha }{a^2}-\frac{\cos \alpha \sin \alpha }{b^2})xy+(\frac{\sin^2 \alpha }{a^2}+\frac{\cos^2 \alpha }{b^2})y^2=1$$
$$(\frac{\cos^2 \alpha }{a^2}+\frac{\sin^2 \alpha }{b^2})x^2+ \sin 2\alpha (\frac{1 }{a^2}-\frac{1}{b^2})xy+(\frac{\sin^2 \alpha }{a^2}+\frac{\cos^2 \alpha }{b^2})y^2=1$$
$$Ax^2+ Bxy+Cy^2=1$$
$x=X-h$
$y=Y-k$
$$A(X-h)^2+ B(X-h)(Y-k)+C(Y-k)^2=1$$
$$A=\frac{\cos^2 \alpha }{a^2}+\frac{\sin^2 \alpha }{b^2}$$
$$B=\sin 2\alpha (\frac{1 }{a^2}-\frac{1}{b^2})$$
$$C=\frac{\sin^2 \alpha }{a^2}+\frac{\cos^2 \alpha }{b^2}$$
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