What are the equations of rotated and shifted ellipse, parabola and hyperbola in the general conic sections form?

Question

How will look the general conic sections equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ in case of rotated and shifted from coordinates origin ellipse, parabola and hyperbola?

I need a formulas for coefficients $A$, $B$, $C$, $D$, $E$ and $F$ for ellipse, hyperbola and parabola. I did it for not-rotated conic sections at the origin of coordinates but have a difficults with rotated and shifted.

For i.e. standart ellipse equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ gives me general equation $\frac{1}{a^2}x^2 + 0xy + \frac{1}{b^2}y^2 + 0x + 0y - 1 = 0$

I need the same in case if ellipse located in position $(h;k)$ and rotated on some angle $\alpha$ from positive $X$ axis. And the same for parabola and hyperbola.

Answer 1

There is a simple way to derive these formulas. Let's take the case of the ellipse. Originally you have $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ If you rotate the coordinate system by an angle $\theta$, you get $$x\to x\cos\theta-y\sin\theta\\y\to x\sin\theta+y\cos\theta$$ So the equation becomes $$\frac{(x\cos\theta-y\sin\theta)^2}{a^2}+\frac{(x\sin\theta+y\cos\theta)^2}{b^2}=1$$ If you translate the origin by $(x_0,y_0)$, the equation above transforms to $$\frac{(x\cos\theta-y\sin\theta+x_0)^2}{a^2}+\frac{(x\sin\theta+y\cos\theta+y_0)^2}{b^2}=1$$ Now all you need to do is to expand the parentheses, and group the terms.

Answer 2

Expand $$(\frac{(x-h)\cos(\alpha)+(y-k)\sin(\alpha)}{a})^2+(\frac{-(x-h)\sin(\alpha)+(y-k)\cos(\alpha)}{b})^2-1=0$$ to get the ellipses. The general equation $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$ might give you an empty conic e.g. expand $$(\frac{(x-h)\cos(\alpha)+(y-k)\sin(\alpha)}{a})^2+(\frac{-(x-h)\sin(\alpha)+(y-k)\cos(\alpha)}{b})^2+1=0.$$ The hyperbolas come from expanding $$(\frac{(x-h)\cos(\alpha)+(y-k)\sin(\alpha)}{a})^2-(\frac{-(x-h)\sin(\alpha)+(y-k)\cos(\alpha)}{b})^2-1=0.$$ The parabolas come from expanding $$2p(-(x-h)\sin(\alpha)+(y-k)\cos(\alpha))= ((x-h)\cos(\alpha)+(y-k)\sin(\alpha))^2,$$ where $p$ is the semi-latus rectum.

And all equations can of course be multiplied by a constant without changing anything.