Conics: Cheat Sheet

Essential Concepts

Ellipses

An ellipse is the set of all points [latex](x,y)[/latex] in a plane such that the sum of their distances from two fixed points (called foci) is constant.

Key features of ellipses:

  • Center: The midpoint of both axes, denoted [latex](h,k)/latex]
  • Vertices: The endpoints of the major axis
  • Co-vertices: The endpoints of the minor axis
  • Foci: Two fixed points that lie on the major axis, related by [latex]c^{2}=a^{2}-b^{2}[/latex]
  • Major axis length: [latex]2a[/latex]
  • Minor axis length: [latex]2b[/latex]

Determining orientation:

  • If [latex]a^{2}[/latex] is under [latex]x^{2}[/latex] (or [latex](x-h)^{2}[/latex]), the major axis is horizontal
  • If [latex]a^{2}[/latex] is under [latex]y^{2}[/latex] (or [latex](y-k)^{2}[/latex]), the major axis is vertical
  • Always [latex]a>b[/latex], where [latex]a[/latex] is associated with the major axis

Hyperbolas

A hyperbola is the set of all points [latex](x,y)[/latex] in a plane such that the difference of the distances from two fixed points (foci) is a positive constant. A hyperbola has two separate unbounded curves that are mirror images of each other.

Key features of hyperbolas:

  • Center: The midpoint of the transverse and conjugate axes
  • Vertices: The endpoints of the transverse axis (the axis containing the foci)
  • Co-vertices: The endpoints of the conjugate axis (perpendicular to transverse axis)
  • Foci: Two fixed points on the transverse axis, related by [latex]c^{2}=a^{2}+b^{2}[/latex]
  • Transverse axis length: [latex]2a[/latex]
  • Conjugate axis length: [latex]2b[/latex]
  • Asymptotes: Lines that the branches approach but never touch

Determining orientation:

  • The positive term determines the transverse axis direction
    • If [latex]x^{2}[/latex] (or [latex](x-h)^{2}[/latex]) is positive, the transverse axis is horizontal
    • If [latex]y^{2}[/latex] (or [latex](y-k)^{2}[/latex]) is positive, the transverse axis is vertical

Parabolas

A parabola is the set of all points [latex](x,y)[/latex] in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Key features of parabolas:

  • Vertex: The point on the parabola closest to the directrix, midway between focus and directrix
  • Focus: A fixed point located [latex]p[/latex] units from the vertex
  • Directrix: A fixed line located [latex]p[/latex] units from the vertex (opposite side from focus)
  • Axis of symmetry: The line through the vertex and focus, perpendicular to the directrix
  • Latus rectum: A line segment through the focus, parallel to the directrix, with endpoints on the parabola; length is [latex]|4p|[/latex]

Determining orientation and direction:

  • If the variable [latex]y[/latex] is squared, the axis of symmetry is horizontal (parabola opens left/right)
  • If the variable [latex]x[/latex] is squared, the axis of symmetry is vertical (parabola opens up/down)

Rotation of Axes

The general form of a conic section is [latex]Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0[/latex], where [latex]A[/latex], [latex]B[/latex], and [latex]C[/latex] are not all zero.

When [latex]B=0[/latex] (non-rotated conics):

  • Ellipse: [latex]A[/latex] and [latex]C[/latex] are nonzero, have the same sign, and [latex]A\neq C[/latex] (equivalently, [latex]AC>0[/latex] and [latex]A\neq C[/latex])
  • Circle: [latex]A[/latex] and [latex]C[/latex] are equal and nonzero ([latex]A=C[/latex])
  • Hyperbola: [latex]A[/latex] and [latex]C[/latex] are nonzero and have opposite signs (equivalently, [latex]AC<0[/latex])
  • Parabola: Either [latex]A[/latex] or [latex]C[/latex] is zero (but not both)

When [latex]B\neq 0[/latex] (rotated conics), use the discriminant [latex]B^{2}-4AC[/latex]:

The discriminant is invariant (remains unchanged after rotation), enabling identification of the conic type:

  • If [latex]B^{2}-4AC<0[/latex], the conic is an ellipse
  • If [latex]B^{2}-4AC=0[/latex], the conic is a parabola
  • If [latex]B^{2}-4AC>0[/latex], the conic is a hyperbola

Rotation of axes:

To eliminate the [latex]xy[/latex] term and rewrite in standard form, rotate the axes by angle [latex]\theta[/latex] where [latex]\cot(2\theta)=\frac{A-C}{B}[/latex]. Use the rotation equations to transform coordinates:

  • [latex]x=x'\cos\theta-y'\sin\theta[/latex]
  • [latex]y=x'\sin\theta+y'\cos\theta[/latex]

After rotation, the angle [latex]\theta[/latex] satisfies:

  • If [latex]\cot(2\theta)>0[/latex], then [latex]0°<\theta<45°[/latex]
  • If [latex]\cot(2\theta)<0[/latex], then [latex]45°<\theta<90°[/latex]
  • If [latex]\cot(2\theta)=0[/latex], then [latex]\theta=45°[/latex]

Degenerate conic sections occur when a plane intersects a double cone through the apex, resulting in a point, a line, or intersecting lines.

Conic Sections in Polar Coordinates

Any conic can be defined by a focus, a directrix, and the eccentricity [latex]e[/latex], which is the ratio [latex]e=\frac{PF}{PD}[/latex], where [latex]PF[/latex] is the distance from a point [latex]P[/latex] on the conic to the focus and [latex]PD[/latex] is the distance from [latex]P[/latex] to the directrix.

Eccentricity determines the conic type:

  • If [latex]0\leq e<1[/latex], the conic is an ellipse
  • If [latex]e=1[/latex], the conic is a parabola
  • If [latex]e>1[/latex], the conic is a hyperbola

Identifying and graphing polar conics:

  1. Rewrite the equation in standard form with 1 in the denominator (multiply numerator and denominator by reciprocal of constant)
  2. Identify [latex]e[/latex] as the coefficient of [latex]\cos\theta[/latex] or [latex]\sin\theta[/latex] in the denominator
  3. Determine the directrix: if [latex]\cos\theta[/latex] appears, directrix is [latex]x=\pm p[/latex]; if [latex]\sin\theta[/latex] appears, directrix is [latex]y=\pm p[/latex]
  4. Set [latex]ep[/latex] equal to the numerator to solve for [latex]p[/latex]
  5. The sign in the denominator indicates directrix position: addition means positive direction, subtraction means negative direction

Converting between polar and rectangular form:

Use the identities [latex]r=\sqrt{x^{2}+y^{2}}[/latex], [latex]x=r\cos\theta[/latex], and [latex]y=r\sin\theta[/latex] to convert equations between coordinate systems.

Key Equations

Horizontal ellipse, center at (h,k) [latex]\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1[/latex], where [latex]a>b[/latex]
Vertical ellipse, center at (h,k) [latex]\frac{(x-h)^{2}}{b^{2}}+\frac{(y-k)^{2}}{a^{2}}=1[/latex], where [latex]a>b[/latex]
Relationship for ellipse foci [latex]c^{2}=a^{2}-b^{2}[/latex]
Hyperbola, center at (h,k), transverse axis parallel to x-axis [latex]\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1[/latex]
Hyperbola, center at (h,k), transverse axis parallel to y-axis [latex]\frac{(y-k)^{2}}{a^{2}}-\frac{(x-h)^{2}}{b^{2}}=1[/latex]
Relationship for hyperbola foci [latex]c^{2}=a^{2}+b^{2}[/latex]
Asymptotes for hyperbola (horizontal transverse axis) [latex]y=\pm\frac{b}{a}(x-h)+k[/latex]
Asymptotes for hyperbola (vertical transverse axis) [latex]y=\pm\frac{a}{b}(x-h)+k[/latex]
Parabola, vertex at (h,k), axis of symmetry parallel to x-axis [latex](y-k)^{2}=4p(x-h)[/latex]
Parabola, vertex at (h,k), axis of symmetry parallel to y-axis [latex](x-h)^{2}=4p(y-k)[/latex]
General form of a conic [latex]Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0[/latex]
Discriminant for identifying conics [latex]B^{2}-4AC[/latex]
Angle of rotation [latex]\cot(2\theta)=\frac{A-C}{B}[/latex]
Rotation of a conic section [latex]x=x'\cos\theta-y'\sin\theta[/latex] and [latex]y=x'\sin\theta+y'\cos\theta[/latex]
Polar equation (directrix x = ±p) [latex]r=\frac{ep}{1\pm e\cos\theta}[/latex]
Polar equation (directrix y = ±p) [latex]r=\frac{ep}{1\pm e\sin\theta}[/latex]

Glossary

angle of rotation

An acute angle [latex]\theta[/latex] formed by a set of axes rotated from the Cartesian plane where: if [latex]\cot(2\theta)>0[/latex], then [latex]\theta[/latex] is between [latex]0°[/latex] and [latex]45°[/latex]; if [latex]\cot(2\theta)<0[/latex], then [latex]\theta[/latex] is between [latex]45°[/latex] and [latex]90°[/latex]; and if [latex]\cot(2\theta)=0[/latex], then [latex]\theta=45°[/latex].

center of an ellipse

The midpoint of both the major and minor axes of an ellipse.

center of a hyperbola

The midpoint of both the transverse and conjugate axes of a hyperbola, where the axes intersect.

central rectangle

A rectangle centered at the hyperbola's center with sides passing through each vertex and co-vertex; the diagonals of this rectangle are the asymptotes of the hyperbola.

conic section (conic)

Any shape resulting from the intersection of a right circular cone with a plane; includes circles, ellipses, parabolas, and hyperbolas.

conjugate axis

The axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its endpoints.

co-vertex

An endpoint of the minor axis of an ellipse or the conjugate axis of a hyperbola.

degenerate conic sections

Any of the possible shapes formed when a plane intersects a double cone through the apex; types include a point, a line, and intersecting lines.

directrix

A fixed line used in the definition of a conic section; for a parabola, all points on the curve are equidistant from the focus and the directrix. More generally, a line such that the ratio of the distance from points on the conic to the focus to the distance to the directrix is constant (the eccentricity).

eccentricity

The ratio of the distances from a point [latex]P[/latex] on the conic to the focus [latex]F[/latex] and to the directrix [latex]D[/latex], represented by [latex]e=\frac{PF}{PD}[/latex], where [latex]e[/latex] is a positive real number. The value of [latex]e[/latex] determines the type of conic.

ellipse

The set of all points [latex](x,y)[/latex] in a plane such that the sum of their distances from two fixed points (foci) is constant.

focus (plural: foci)

A fixed point used in the definition of a conic section. For an ellipse, one of two fixed points on the major axis such that the sum of distances from these points to any point on the ellipse is constant. For a hyperbola, one of two fixed points such that the difference of distances is constant. For a parabola, a fixed point in the interior that lies on the axis of symmetry.

hyperbola

The set of all points [latex](x,y)[/latex] in a plane such that the difference of the distances between latex[/latex] and two fixed points (foci) is a positive constant.

invariant

A property or expression that remains unchanged after a transformation, such as rotation. The discriminant [latex]B^{2}-4AC[/latex] is invariant under rotation.

latus rectum

The line segment that passes through the focus of a parabola, parallel to the directrix, with endpoints on the parabola; its length is [latex]|4p|[/latex].

major axis

The longer of the two axes of an ellipse that passes through the center, both foci, and both vertices.

minor axis

The shorter of the two axes of an ellipse that passes through the center and both co-vertices, perpendicular to the major axis.

nondegenerate conic section

A shape formed by the intersection of a plane with a double right cone such that the plane does not pass through the apex; nondegenerate conics include circles, ellipses, hyperbolas, and parabolas.

parabola

The set of all points [latex](x,y)[/latex] in a plane that are the same distance from a fixed point (the focus) and a fixed line (the directrix).

polar equation

An equation of a curve in polar coordinates [latex]r[/latex] and [latex]\theta[/latex].

transverse axis

The axis of a hyperbola that passes through the center and has the vertices as its endpoints; the foci lie on this axis.

vertex (plural: vertices)

For an ellipse, an endpoint of the major axis; for a hyperbola, an endpoint of the transverse axis; for a parabola, the point on the curve closest to the directrix and midway between the focus and directrix.