Ellipse with Shifted Centre

Centre at (h, k). All elements of the standard ellipse apply with origin shifted to (h, k). Identified by completing the square in the general second-degree equation.

When the centre is at $(h, k)$, substitute $X = x - h$, $Y = y - k$. In $(X, Y)$ coordinates the ellipse is $X^2/a^2 + Y^2/b^2 = 1$, which translates back to:

$$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$

Identifying from the general equation: Any equation $Ax^2 + Cy^2 + Dx + Ey + F = 0$ (with $A, C > 0$, $A \neq C$, no $xy$ term) represents an ellipse. Complete the square in both $x$ and $y$:

$$A(x-h)^2 + C(y-k)^2 = K$$

Divide by $K$ to bring to standard form, with $a^2 = K/A$ and $b^2 = K/C$.

Elements in original coordinates for $(x-h)^2/a^2 + (y-k)^2/b^2 = 1$ with $a > b$: