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Conics

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

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  1. Conics Advanced Math Section 4.3

  2. Conic • AKA conic section • Intersection of a plane and a double-napped cone • See figure 4.18 on page 354 Advanced Math 4.3

  3. Degenerate conic • Plane passes through vertex of the cone • See figure 4.19 on page 354 Advanced Math 4.3

  4. 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

  5. Circle • Section 1.1 • The collection of all points (x, y) that are equidistant from a fixed point (h, k). Advanced Math 4.3

  6. 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

  7. 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

  8. Examples • Find the focus and directrix of each parabola Advanced Math 4.3

  9. 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

  10. 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

  11. Example • Find the center and vertices of the following ellipse and sketch its graph Advanced Math 4.3

  12. 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

  13. 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

  14. 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

  15. 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

  16. Example • Sketch the graph of the following hyperbola Advanced Math 4.3

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