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Conics. Advanced Math Section 4.3. Conic. AKA conic section Intersection of a plane and a double-napped cone See figure 4.18 on page 354. Degenerate conic. Plane passes through vertex of the cone See figure 4.19 on page 354. Three ways to approach conics.

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Conics

Conics

Advanced Math

Section 4.3


Conic
Conic

  • AKA conic section

  • Intersection of a plane and a double-napped cone

  • See figure 4.18 on page 354

Advanced Math 4.3


Degenerate conic
Degenerate conic

  • Plane passes through vertex of the cone

  • See figure 4.19 on page 354

Advanced Math 4.3


Three ways to approach conics
Three ways to approach conics

  • Intersections of planes and cones

    • Original Greeks

  • Algebraically

    • General second-degree equation

      • Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

  • Locus (collection) of points satisfying a general property

    • What we’ll use

Advanced Math 4.3


Circle
Circle

  • Section 1.1

  • The collection of all points (x, y) that are equidistant from a fixed point (h, k).

Advanced Math 4.3


Parabola
Parabola

  • Set of all points (x, y) in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus, not on the line. (see figure 4.20 on page 355)

  • The vertex is the midpoint between the focus and the directrix.

  • The axis of the parabola is the line passing through the focus and the vertex.

    • Can be vertical or horizontal

    • Parabola is symmetric with respect to its axis

Advanced Math 4.3


Standard equation of a parabola
Standard equation of a parabola

  • (Vertex at origin) see page 355

  • The focus is on the axis p units (directed distance) from the vertex

  • Focus is (0, p) for vertical axis

  • Focus is (p, 0) for horizontal axis

Advanced Math 4.3


Examples
Examples

  • Find the focus and directrix of each parabola

Advanced Math 4.3


Ellipse
Ellipse

  • Set of all points (x, y) in a plane the sum of whose distances from two distinct points (foci) is constant. (See figure 4.25 on page 357)

  • A line through the foci intersects the ellipse at two vertices.

  • The major axis connects the two vertices

  • The center is the midpoint of the major axis

  • The minor axis is perpendicular to the major axis at the center

Advanced Math 4.3


Standard equation of an ellipse
Standard equation of an ellipse

  • (center at origin) see page 357

  • Vertices lie on major axis a units from center

  • Foci lie on major axis c units from center

Advanced Math 4.3


Example
Example

  • Find the center and vertices of the following ellipse and sketch its graph

Advanced Math 4.3


Hyperbola
Hyperbola

  • Set of all points (x, y) in a plane the difference of whose distances from two distinct points (foci) is a positive constant (see figure 4.30 on page 359)

  • Graph has two disconnected branches

  • The line through the foci intersects the hyperbola at two vertices

  • The transverse axis connects the vertices

  • The center is the midpoint of the transverse axis.

Advanced Math 4.3


Standard equation of a hyperbola
Standard equation of a hyperbola

  • (center at origin) see page 359

  • Vertices lie on transverse axis a units from center

  • Foci lie on transverse axis c units from center

Advanced Math 4.3


Example1
Example

  • Find the standard form of the equation of a hyperbola with center at the origin, vertices (0, 2) and (0, -2), and foci (0, -3) and (0, 3).

Advanced Math 4.3


Asymptotes of a hyperbola
Asymptotes of a hyperbola

  • (center at origin)

  • Useful for graphing

  • Pass through the corners of a rectangle of dimensions 2a by 2b.

  • The conjugate axis has length 2b and joins either (0, b) with (0, -b) or (b, 0) with (-b, 0)

Advanced Math 4.3


Example2
Example

  • Sketch the graph of the following hyperbola

Advanced Math 4.3


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