What are the equations of rotated and shifted ellipse, parabola and hyperbola in the general conic s

Kale Bright

Kale Bright

Answered question

2022-04-18

What are the equations of rotated and shifted ellipse, parabola and hyperbola in the general conic sections form?
How will look the general conic sections equation Ax2+Bxy+Cy2+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 x2a2+y2b2=1 gives me general equation 1a2x2+0xy+1b2y2+0x+0y1=0
I need the same in case if ellipse located in position (h;k) and rotated on some angle α from positive X axis. And the same for parabola and hyperbola.

Answer & Explanation

ysnlm8eut

ysnlm8eut

Beginner2022-04-19Added 14 answers

There is a simple way to derive these formulas. Let's take the case of the ellipse. Originally you have x2a2+y2b2=1
If you rotate the coordinate system by an angle θ, you get
xxcosθysinθ

yxsinθ+ycosθ
So the equation becomes
(xcosθysinθ)2a2+(xsinθ+ycosθ)2b2=1
If you translate the origin by (x0,y0), the equation above transforms to
(xcosθysinθ+x0)2a2+(xsinθ+ycosθ+y0)2b2=1
Now all you need to do is to expand the parentheses, and group the terms.

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