Information Section:

Definition Section:

Overview: Conic Sections - where the name came from and how do we dervie parabolas, ellipses and hyperbolas from the conic sections.

Conic Section Cone - Figure formed by rotating one of two intersecting lines about the other.

 

Axis of the cone - The fixed line of the two used to form the cone.

 

Angle q - The angle between the Axis of the cone and the line rotating about the axis.

 

Edge of the cone - the surface formed by the line rotating around the axis of the cone.

 

Vertex - The point of intersection where the two lines meet.

 

Nappe - the section of the cone either above or below the vertex. These are referred to as upper or lower nappes.

 

Parabola - Figure formed when a plane intersects the cone parallel to the to the edge of the nappe (and cuts through either only the upper or the lower nappe.  Locus (or set) of all points whose distance from a fixed point (focus) equal their distance from a fixed line (directorix).

 

Ellipse - Figure formed when a plane intersects  a single nappe of the cone, other than parallel.  Locus of points such that the sum of the distances of each point from two given points (the foci) is a constant.

 

Hyperbola - Figure formed when a plane intersects both nappes of the cone. Locus of points such that the difference between the distances of each point from two given points (the foci) is a constant.

 

Parabola: Definitions of parabola, locus, equidistant, directorix, focus and vertex. Basic translation of the parabolic equation. Theorems concerning translation and the parabola Corollary to the translation theorem. Vertex form of the parabolic equation (vs. the full form)

Locus - The set of all points equidistant from both the directorix and the focus.

 

Equidistant - The same distance between items.

 

Directorix - The fixed line in the construction of a parabola.

 

Focus - The fixed point in the construction of a parabola.

 

Vertex - A point constructed via a perpendicular line from the focus, directly to the directorix.  It is the point of intersection of the locus and perpendicular line.  This perpendicular line is also considered the axis of symmetry.

 

Translation - The movement of a figure (e.g. parabola) either left/right or up/down from the initial location of the figure.

 

Vertex form of an equation for a parabola -     y - k = a(x - h)^2

 

Ellipse: Definitions; PF1 + PF2 = k; the basic theorem concerning the ellipse; Equations

Foci - The two fixed points in the construction of an ellipse.

 

Ellipse Definition - Let F1 and F2 be two given points in a plane, and k be a positive real number with k > F1F2.  Then the ellipse with foci F1 and F2 and focal constant k is the set of all points in the planes which satisfy PF1 + PF2 = k.

 

Length of the horizontal axis for ellipse with equation x^2/a^2 + y^2/b^2 = 1; is  2a

 

Length of the vertical axis for ellipse with equation x^2/a^2 + y^2/b^2 = 1; is 2b

 

Distance between the foci, where c^2 = a^2 - b^2, for ellipse with equation x^2/a^2 - y^2/b^2 = 1; is  2c

 

Hyperbola: Dandelin and the spheres... Definitions of asymptote, focal length etc. Theorems (|PF1 - PF2| = k); Corollaries; Equations

Hyperbola Definition - Let F1 and F2 be two given points in a plane, and k be a positive real number with k < F1F2.  Then the hyperbola with foci F1 and F2 and focal constant k is the set of all points in the planes which satisfy |PF1 - PF2| = k.

 

Length of the horizontal axis for hyperbola with equation x^2/a^2 - y^2/b^2 = 1; is  2a

 

Length of the vertical axis for hyperbola with equation x^2/a^2 - y^2/b^2 = 1; is 2b

 

Distance between the foci, where c^2 = a^2 + b^2, for hyperbola with equation x^2/a^2 - y^2/b^2 = 1; is  2c

 

Isometry - A composite of translations, rotations and reflections.

 

Rectangular Hyperbola - An hyperbola where the asymptotes are perpendicular.

 

Asymptote of the hyperbola - Lines that the hyperbola approaches, as x gets further from the foci.

 

Vertices of the hyperbola - With the hyperbola centered about the origin, the points (-a, 0) and (a, 0).  Points of closest distance between the two parts of the hyperbola.

 

Axis of the hyperbola - two lines of symmetry for the hyperbola;  intersect at the center of the hyperbola.

 

Center of the hyperbola - midpoint between the two foci of the hyperbola.  Also is the intersection of the asymptotes of the hyperbola.

Theorems and Corollaries Section:

Graph-Translation Theorem - In a sentence for a graph, replacing x by x - h and y by y - k causes the graph to undergo the translation Th,k

Graph-Translation Corollary - The image of the parabola y = ax^2 under the translation Th,k is  y - k = a(x - h)^2

Ellipse Theorem - An ellipse with foci F1 and F2 is reflection symmetric to line F1F2 and to the perpendicular bisector of the segment F1F2.

Equation for Ellipse Theorem - The hyperbola with foci (c, 0) and (-c, 0) and focal constant 2a has equation x^2/a^2 + y^2/b^2 = 1, where c^2 = a^2 - b^2.

Hyperbola Theorem - A hyperbola with foci F1 and F2 is reflection symmetric to line F1F2 and to the perpendicular bisector of the segment F1F2.

Equation for Hyperbola Theorem - The hyperbola with foci (c, 0) and (-c, 0) and focal constant 2a has equation x^2/a^2 - y^2/b^2 = 1, where b^2 = c^2 - a^2.

Graph-Rotation Theorem - In a relation described by a sentence in x and y, the following two processes yield the same graph:

1.  Replacing x by x cos q + y sin q  and y by -x sin q + y cos q

2.  Applying the rotation of magnitude q about the origin to the graph of the original equation.

Investigation Section

Activity 1 - Review of Parabolic equations

Activity 2 - Review of equations for ellipse

Activity 3 - Review of equations for hyperbola

Activity 4 - Review of Rotations and Translations.

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