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# Conics - PowerPoint PPT Presentation

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

Section 4.3

• AKA conic section

• Intersection of a plane and a double-napped cone

• See figure 4.18 on page 354

• Plane passes through vertex of the cone

• See figure 4.19 on page 354

• 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

• Section 1.1

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

• 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

• (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

• Find the focus and directrix of each parabola

• 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

• (center at origin) see page 357

• Vertices lie on major axis a units from center

• Foci lie on major axis c units from center

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

• 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.

• (center at origin) see page 359

• Vertices lie on transverse axis a units from center

• Foci lie on transverse axis c units from center

• 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).

• (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)