Shifted Conics and Completing the Square

Definition Let $h$ and $k$ be real numbers. Moving every point of the plane $h$ units horizontally and $k$ units vertically sends $(x, y)$ to $(x + h, y + k)$. If a curve is the solution set of an equation $G(x, y) = 0$, the moved curve is the solution set of

because $(x, y)$ lies on the moved curve exactly when $(x - h, y - k)$ lies on the original one. A translation keeps all distances, so a translated conic has the same type, size, shape and eccentricity as the original.

Definition Let $p \ne 0$. The parabola with vertex $(h, k)$, focus $(h + p, k)$ and directrix $x = h - p$ has the equation

Its axis is the horizontal line $y = k$, and it opens to the right when $p > 0$ and to the left when $p < 0$. The parabola with vertex $(h, k)$, focus $(h, k + p)$ and directrix $y = k - p$ has the equation

Its axis is the vertical line $x = h$, and it opens upward when $p > 0$ and downward when $p < 0$.

Definition Let $a > b > 0$ and $c = \sqrt{a^2 - b^2}$. The ellipse with center $(h, k)$ and a horizontal major axis is

Its vertices are $(h \pm a, k)$, its covertices are $(h, k \pm b)$ and its foci are $(h \pm c, k)$. When $a^2$ stands under $(y - k)^2$ instead, the major axis is vertical, the vertices are $(h, k \pm a)$, the covertices are $(h \pm b, k)$ and the foci are $(h, k \pm c)$. If both denominators equal $a^2$, the equation describes the circle of radius $a$ about $(h, k)$ instead.

Definition Let $a > 0$, $b > 0$ and $c = \sqrt{a^2 + b^2}$. The hyperbola with center $(h, k)$ and a horizontal transverse axis is

with vertices $(h \pm a, k)$, foci $(h \pm c, k)$ and asymptotes $y - k = \pm \frac{b}{a}(x - h)$. The hyperbola with center $(h, k)$ and a vertical transverse axis is

with vertices $(h, k \pm a)$, foci $(h, k \pm c)$ and asymptotes $y - k = \pm \frac{a}{b}(x - h)$.

Definition For real numbers $A \ne 0$ and $D$,

An equation $Ax^2 + Cy^2 + Dx + Ey + F = 0$ with no $xy$ term, where $A$ and $C$ are not both zero, is rewritten in shifted form this way. Group the $x$ terms and the $y$ terms, and factor out the coefficient of each squared variable. Complete the square inside each group that has a square, and add to the other side exactly what this adds to the left side. If both variables are squared and the new right side is not zero, divide by it so that the right side is $1$. If only one variable is squared and the other appears to the first power, move the remaining terms to the right side and factor out the coefficient of that variable, as in the parabola forms above. Some outcomes are degenerate, and the last section shows how to recognize them.

Definition The translation by $h$ horizontally and $k$ vertically moves every feature of a conic by the same amount. The center, the vertices, the covertices and the foci move from $(x, y)$ to $(x + h, y + k)$, and the axes, directrices and asymptotes move to parallel lines through the moved points. The numbers $a$, $b$, $c$, $p$ and $e$ do not change. So each point feature of a shifted conic is the matching point of the unshifted conic plus $(h, k)$. In the other direction, the equation of an ellipse or a hyperbola can be found from shifted features by locating the center first, as the midpoint of the vertices or of the foci, and then reading $a$, $b$ and $c$ as distances from it.

Definition Consider $Ax^2 + Cy^2 + Dx + Ey + F = 0$ with $A$ and $C$ not both zero.

• If $AC > 0$, the graph is an ellipse (a circle when $A = C$), a single point, or empty.
• If $AC < 0$, the graph is a hyperbola or a pair of lines crossing at one point.
• If exactly one of $A$ and $C$ is zero, the graph is a parabola, a pair of parallel lines, a single line, or empty.

Completing the square decides which case happens. When $AC \ne 0$ the result has the form $A(x - h)^2 + C(y - k)^2 = K$, and the graph is an ellipse or a hyperbola exactly when $K \ne 0$ and, in the case $AC > 0$, $K$ has the same sign as $A$.