HPC 8.1-8.3: Conic Sections (Parabolas, Ellipses, Hyperbolas)

Conic Sections defined algebraically in the Cartesian plane

Ax²+Bxy+Cy²+Dx+Ey+F=0

Parabola (def, terms, standard form for vertex (h, k))

Def: the set of all points in the plane that are simultaneously equidistant from the focus and the directrix

Terms:

• Axis of the parabola/ line/axis of symmetry: line that passes through the focus and is perpendicular to the directrix

• Vertex: point where the axis of the parabola intersects the parabola itself (midway between the focus and directrix)

• Focal length (p): distance from the vertex to the focus/directrix

• Focal chord: any segment with endpoints on the parabola that passes through the focus

• Latus rectum: focal chord perpendicular to the axis of the parabola (length = focal width = 4p)

  • Endpoints for focus point (x, y) if parabola is:

    • Vertical: (x±2p, y)

    • Horizontal: (x, y±2p)

Standard form if opening where p=focal length:

• Up: (x-h)² = 4p(y-k)

• Down: (x-h)² = -4p(y-k)

• Right: (y-h)² = 4p(x-k)

• Left: (y-h)² = -4p(x-k)

Translating parabola into standard form from algebraic form (Ax²+Bx+Cy+D=0 or switch x and y)

1. Isolate second degree variable and complete square

2. Factor out coefficient of other variable

Finding standard form of parabola from vertex and point

plug both in to solve for p

Name and properties of parabola rotated about its axis in 3D

Paraboloid of revolution; light parallel to axis of parabola is reflected towards the focus and visa versa (e.g. flashlight goes from focus out and satellite dish takes in signals to focus)

Ellipse (def, terms, standard form, graphing)

Def: given two points in a plane, it is the set of all points in the plane such that the sum of the distances to these points is constant

Terms:

• Foci

• Center

• Major axis (always longer than minor)

• Minor axis

• a = semimajor axis

• b = semiminor axis

• c = distance from center to foci

• a²=b²+c²

• Vertices (major vertices): points where the ellipse intersects the focal axis

• eccentricity (e) = c/a (0<e<1)

  • If e=0: circle

  • If e=1: parabola (not line)

Standard from for center (h, k):

• For horizontal focal axis: (x-h)²/a²+(y-k)²/b² = 1

• Switch a and b for vertical

Graphing: center, foci, vertices, and minor axis endpoints

Translating ellipse into standard form from algebraic form (Ax²+By²+Cx+Dy+E=0)

Complete squares

Parametric equations for ellipse

x(t) = a cost + h

y(t) = b sint + k

0≤t≤2π

or a and b switch for vertical ellipse

Planet lingo

(In their elliptical orbits)

Perihelion: closest point to sun

Aphelion: furthest point from sun

Name and properties of ellipse rotated about its focal axis in 3D

Ellipsoid of revolution; signal/light sent from one focal point is reflected off ellipsoid toward other focal point

Hyperbola (def, terms, standard form for center (h, k), asymptotes)

Def: given two fixed points (foci) in a plane, it is the set of all points in the plane such that the difference of the distances to these fixed points is constant (2a)

Terms:

• focal axis

• center

• vertices (on focal axis) (h±a, k)

• c²=a²+b²

• a = distance to vertex

• c = distance to foci

• e = c/a (e>1)

• Transverse axis: connects the vertices (=2a)

• Conjugate axis: perpendicular to transverse axis w/ midpoint at center (=2b)

• Focal width = 2b²/a

Standard form for:

• Horizontal: (x-h)²/a² - (y-k)²/b² = 1

• Vertical: (y-k)²/a² - (x-h)²/b² = 1

• a² is always under the positive term but is not always larger than b²

Asymptotes:

• Horizontal: m=±b/a

• Vertical: m=±a/b

• Contain the center point - plug and chug

Translating hyperbola into standard form from algebraic form (Ax²-By²+Cx+Dy+E=0 for horizontal or -Ax²+By² for vertical)

complete square

Parametric equations for a hyperbola

x(t) = a sect + h

y(t) = b tant + k

-π/2<t<3π/2, t≠π/2

variable associated with a gives direction of focal (and transverse) axis; asect and btant stay together

Degenerate hyperbolas (form, appearance/graph)

(x-h)²/a² - (y-k)²/b² = 0 or (y-k)²/a² - (x-h)²/b² = 1

if x is over a in standard parabola form: y-k=±b/a(x-h)

otherwise: y-k=±a/b(x-h)

Conjugate hyperbolas

x and y switch