Translated ellipses

Not all ellipses have their centre at the origin. If the centre shifts but the axes stay parallel to the Cartesian axes, the form of the equation changes in a predictable way.

If the ellipse has centre $O'(x_0;y_0)$ but axes parallel to the Cartesian axes, the canonical form becomes $\frac{(x-x_0)^2}{a^2} + \frac{(y-y_0)^2}{b^2} = 1.$

Starting from an equation in expanded form of the type $A x^2 + B y^2 + Cx + Dy + E = 0$ (with $A, B$ positive and of the same sign, but $A \ne B$), one recognises it and brings it into translated canonical form by completing the square, exactly as is done to recognise a parabola from its expanded form.

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Topics: Ellipse
Concepts: Completing the square · Ellipse · Translated ellipse · Canonical equation
Skills: Analytic geometry · Using formulae