MTH 253 Calculus (Other Topics) Chapter 10 – Conic Sections and Polar Coordinates Section 10.1 – Conic Sections and Quadratic Equations Copyright © 2009 by Ron Wallace, all rights reserved.

Conic Sections The intersection of a right circular cone and a plane will produce one of four curves: Circle Ellipse Parabola Hyperbola

Quadratic Equation (2 variables) It can, and will, be shown that any quadratic equation in two variables will be a conic section (and vice versa). Begin with curves “centered” at the origin Move the center Rotate (B ≠ 0)

Parabola  Definition The set of all points equidistant from a fixed point (focus) and a line (directrix). The intersection of a right circular cone and a plane that is parallel to an element of the cone. directrix focus point parabola

Parabola – Simple Cases  Focus: (0, p) Directrix: y = –p (0,p) (x,y) (x,–p) 1 of 4 Note that the points (2p,p) lie on this curve.

Parabola - Examples  Graph the following …

Parabola – Simple Cases  Focus: (0, –p) Directrix: y = p (0,–p) (x,y) (x,p) 2 of 4 Note that the points (2p,–p) lie on this curve.

Parabola – Simple Cases  Focus: (p, 0) Directrix: x = –p (p,0) (x,y) (–p,y) 3 of 4 Note that the points (p,2p) lie on this curve.

Parabola – Simple Cases  Focus: (–p, 0) Directrix: x = p (–p,0) (x,y) (p,y) 4 of 4 Note that the points (–p,2p) lie on this curve.

Parabola  The point halfway between the focus and directrix is called the vertex. w/ the 4 simple cases, the origin  How does the value of p affect the parabola? Small p  narrow Large p  wide

Ellipse  Definition The set of all points whose sum of distances to two fixed points (foci) is a constant. The intersection of a right circular cone and a plane that cuts clear through one of the nappes of the cone.

Ellipse  Terms Focal axis: line containing the foci  aka: major axis Center: point halfway between the foci Vertices: intersection of the ellipse with the focal axis

Ellipse – Simple Cases  Foci: (c, 0) Vertices: (a, 0) Sum = 2a (a,0) (x,y) (–a,0) 1 of 2 (c,0)(–c,0) (0,b) (0,–b) Note that a > c & a > b.

Ellipse - Examples  Graph the following …

Ellipse – Simple Cases  Foci: (0, c) Vertices: (0, a) Sum = 2a (b,0) (x,y) (–b,0) 2 of 2 (0,c) (0,–c) (0,a) (0,–a) Note that a > c & a > b.

Ellipse  How does the difference between a & c affect the shape of the ellipse? a–c small  oval a–c large  round  What happens if c = 0? The foci and the center are all the same The result is a circle (b = a = r) The intersection of a cone with a plane that is perpendicular to the axis of the cone.

Hyperbola  Definition The set of all points whose difference of distances to two fixed points (foci) is a constant. The intersection of a right circular cone and a plane that cuts intersects both nappes of the cone.

Hyperbola  Terms Focal axis: line containing the foci Center: point halfway between the foci Vertices: intersection of the hyperbola with the focal axis

Hyperbola – Simple Cases  Foci: (c, 0) Vertices: (a, 0) Difference = 2a (a,0) (x,y) (–a,0) 1 of 2 (c,0)(–c,0) Note that c > a & c > b.

Hyperbola – Simple Cases  Asymptotes? (a,0) (x,y) (–a,0) 1 of 2 (c,0)(–c,0) (0,b) (0,–b) Note that c > a & c > b.

Hyperbola - Examples  Graph the following …

Hyperbola – Simple Cases  Foci: (0, c) Vertices: (0, a) Difference = 2a (a,0) (x,y) (–a,0) 2 of 2 (c,0) (–c,0) Note that c > a & c > b. (b,0) (–b,0) Asymptotes?

Reflective Properties of Conics  Parabola: Rays perpendicular to the directrix will reflect to the focus  Ellipse: Rays emitting from one focus will reflect to the other focus.  Hyperbola: Rays moving towards one focus will reflect to the other focus. See diagrams of applications on page 677.

Shifting Conics  To move the center of a conic to the point (h,k), replace x with x–h and replace y with y–k.

Conics & Quadratic Equations  Complete the squares and simplify.  Parabola: A=0 or C=0  Ellipse: A & C have the same signs  Hyperbola: A & C have opposite signs

Conics & Quadratic Equations  Examples