# Ellipse general equation from dimensions, offset, and tilt angle

Given an ellipse with the following parameters:

• $$a$$ = semimajor axis
• $$b$$ = semiminor axis
• $$\theta$$ = tilt angle from horizontal
• $$(\Delta x, \Delta y)$$ = position of the center

How do I find the general formula of that ellipse, namely in the form

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

I’ve looked everywhere, and I can’t find anything even close to that.

$$\frac {((x-\Delta x)\cos\theta + (y-\Delta y)\sin \theta)^2}{a^2} + \frac {(-(x-\Delta x)\sin\theta + (y-\Delta y)\cos\theta)^2}{b^2} = 1$$