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## PowerPoint Slideshow about 'Conics' - zasha

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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

- Plane passes through vertex of the cone
- See figure 4.19 on page 354

Advanced Math 4.3

Three ways to approach conics

- Intersections of planes and cones
- Original Greeks

- Algebraically
- General second-degree equation
- Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

- General second-degree equation
- Locus (collection) of points satisfying a general property
- What we’ll use

Advanced Math 4.3

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

- 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

- (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

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

- (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

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

Advanced Math 4.3

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

- (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

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

- (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

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