ACT Math 6: Coordinate Geometry — Conic Sections and Transformations

Conic section

A curve formed by intersecting a plane with the surface of a double cone. The nondegenerate conics are ellipses, parabolas, and hyperbolas; a circle is a special ellipse.

How does the orientation of a plane cutting a double cone determine the type of conic?

A closed intersection is an ellipse, a plane parallel to one generating line produces a parabola, and a plane intersecting both nappes produces a hyperbola. A circle occurs when the plane is perpendicular to the axis of a right circular cone.

Degenerate conic

A conic produced when the cutting plane passes through the cone's vertex. Depending on the plane, the result can be a point, a line, or a pair of intersecting lines.

Focus and directrix

A focus is a fixed point and a directrix is a fixed line used to define a conic. A point $P$ lies on the conic when its distance to the focus is a fixed multiple of its perpendicular distance to the directrix.

Focus-directrix definition of eccentricity

For a conic, eccentricity is $e = PF/PD$, where $F$ is the focus and $D$ is the foot of the perpendicular from $P$ to the directrix. It is constant for every point on the conic.

How does eccentricity classify noncircular conics?

If $0 < e < 1$, the conic is an ellipse; if $e = 1$, it is a parabola; and if $e > 1$, it is a hyperbola. A circle has eccentricity $e=0$ as a limiting case.

Two-focus definition of an ellipse

An ellipse is the locus of points for which the sum of the distances to two fixed foci is constant: $PF_1+PF_2=2a$, where $a$ is the semi-major axis.

Two-focus definition of a hyperbola

A hyperbola is the locus of points for which the absolute difference of the distances to two fixed foci is constant: $|PF_1-PF_2|=2a$.

Principal axis

The principal axis is the line joining the foci of an ellipse or hyperbola. Its midpoint is the center of the conic; a parabola has no center.

Standard equation of a circle centered at the origin

A circle with radius $a$ has equation $x^2+y^2=a^2$. It is an ellipse with equal semi-axes, so $a=b$ and $c=0$.

Standard equation of an ellipse centered at the origin with major axis on the $x$-axis

The equation is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, with $a\ge b>0$. Its vertices are $(\pm a,0)$, foci are $(\pm c,0)$, and $c^2=a^2-b^2$.

Standard equation of an ellipse centered at the origin with major axis on the $y$-axis

The equation is $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$, with $a\ge b>0$. Its vertices are $(0,\pm a)$, foci are $(0,\pm c)$, and $c^2=a^2-b^2$.

Standard equation of a hyperbola centered at the origin with transverse axis on the $x$-axis

The equation is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. Its vertices are $(\pm a,0)$, foci are $(\pm c,0)$, and $c^2=a^2+b^2$.

Standard equation of a hyperbola centered at the origin with transverse axis on the $y$-axis

The equation is $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$. Its vertices are $(0,\pm a)$, foci are $(0,\pm c)$, and $c^2=a^2+b^2$.

Standard equation of a right-opening parabola

The equation is $y^2=4ax$ with $a>0$. Its vertex is $(0,0)$, focus is $(a,0)$, directrix is $x=-a$, and its latus rectum has length $4a$.

Standard equation of an upward-opening parabola

The equation is $x^2=4ay$ with $a>0$. Its vertex is $(0,0)$, focus is $(0,a)$, directrix is $y=-a$, and its axis of symmetry is the $y$-axis.

How do translations and rotations affect conic equations?

A suitable translation moves the conic's center or vertex to the origin, and a rotation can eliminate an $xy$ term and align the principal axis with a coordinate axis. These coordinate changes convert many conics to standard form.

Symmetries of the standard conic equations

The standard circle, ellipse, and horizontal hyperbola are symmetric about both coordinate axes. The standard parabola $y^2=4ax$ is symmetric about the $x$-axis, while the rectangular hyperbola $xy=c^2$ is symmetric about $y=x$ and $y=-x$.

Standard equation of a rectangular hyperbola

A rectangular, or equilateral, hyperbola can be written as $xy=c^2$. Its asymptotes are the coordinate axes and are perpendicular.

General Cartesian equation of a conic

Every plane quadratic curve can be written as $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$, where $A$, $B$, and $C$ are not all zero. The equation may represent a nondegenerate or degenerate conic.

Discriminant classification of a general conic

For $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$, the discriminant is $B^2-4AC$. For a nondegenerate conic, it is negative for an ellipse, zero for a parabola, and positive for a hyperbola.

How can a circle be recognized from the general quadratic equation?

Within the ellipse case, the equation represents a circle when $A=C$ and $B=0$. This means the quadratic terms have equal coefficients and no rotation-producing $xy$ term.

Translated circle equation

A circle with center $(h,k)$ and radius $r$ has equation $(x-h)^2+(y-k)^2=r^2$.

Translated ellipse equation

An axis-aligned ellipse with center $(h,k)$ has equation $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$. The values $a$ and $b$ are the semi-axis lengths.

Translated hyperbola equation

An axis-aligned hyperbola with center $(h,k)$ and horizontal transverse axis has equation $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$. Its asymptotes are $y-k=\pm\frac{b}{a}(x-h)$.

Translated parabola equations

A horizontal parabola has equation $(y-k)^2=4a(x-h)$, and a vertical parabola has equation $(x-h)^2=4a(y-k)$. In either case, $(h,k)$ is the vertex.