Translated conics and converting from general form

Translating a conic

• Any conic can be moved (translated) so its centre or vertex sits at $(h, k)$ instead of the origin.
• Replace $x$ with $(x - h)$ and $y$ with $(y - k)$ in the standard equation.
• The shape and size stay the same — only the position changes.

The translated standard forms

• Circle: $(x - h)^2 + (y - k)^2 = r^2$.
• Ellipse: $\dfrac{(x - h)^2}{a^2} + \dfrac{(y - k)^2}{b^2} = 1$.
• Hyperbola: $\dfrac{(x - h)^2}{a^2} - \dfrac{(y - k)^2}{b^2} = 1$.
• Parabola: $(y - k)^2 = 4a(x - h)$ (opening right from vertex $(h, k)$).
• In every case the centre or vertex is at $(h, k)$.

Locating features after a translation

• Find the feature at the origin first (centre, foci, vertices, directrix), using $a$, $b$, $r$.
• Then shift each feature by $(+h, +k)$.
• Example: if a focus sits at $(c, 0)$ before translating, after translating it is at $(h + c, k)$.

Converting general form to standard form

• A translated conic is often expanded into a general second-degree equation.
• Group the $x$-terms and the $y$-terms.
• Complete the square in $x$ and in $y$ (the same method as for the circle).
• Divide through if needed so the right-hand side is $1$ (for an ellipse or hyperbola).