Conic Sections: Introduction Circle Parabola Ellipse Studied since 4th Cent BC Circles basic shape for wheels, gears used in machinery Parabolas describe path of projectile; in business a model for max/min; in physics a model for parabolic reflectors & antenna Ellipse important in astronomy. In 1609, Kepler (German)proved that planents moved in elliptical orbits. Hyperbola describes shock wave of sonic boom. Systems of electronic navigation based on hyperbola Hyperbola Antimated Conic created by Preston Nichols. See credits last page. ©2002 Roy L. Gover (www.mrgover.com)

Definition Conic sections are the figures obtained by slicing a right circular cone at various angles Idea of conics was first developed by Appolonious (3 B.C.), a Greek geometer

Definition The study of conic sections is part of a broader subject called analytic geometry: the combination of algebra and geometry.

Important Idea Equations for the conic sections are quadratics of the general form: Where A, B, D, E & F are constants.

Prerequisite Knowledge In solving conic section problems, we need to review: Distance formula Completing the square

Example Distance Formula:

Example Find the distance from (-5,3) to (4,-6)

Example x2 & x1 x2 x1 y1 y2 1. Choose y2 & y1 Find the distance from (-5,3) to (4,-6) 2. Select corresponding x2 & x1 x2 x1 y1 y2 3. Substitute in distance formula

Example Find the distance from (-5,3) to (4,-6) x2 x1 y1 y2

Try This Find the distance from (6,2) to (-4,-6)

Example Complete the square for: 1. Half the coefficient of x: 1/2 of 12=6 2. Square this number and add to the expression

Important Idea Completing the square is the process of finding the number that will make the expression a perfect square trinomial.

Important Idea is a perfect square trinomial because it factors as:

Try This Complete the square for: then factor your result

Example Complete the square for: then factor your result

Example Solve by completing the square: 3. Factor the left side and solve: 1. Move the constant to the right: 2. Complete the square and add to the left and right:

Try This Solve by completing the square:

Example Solve by completing the square… Before you complete the square, the coefficient of the squared term must be 1

Try This Solve by completing the square…

Lesson Close You will need to complete the square to solve some of the conic section problems in this chapter. Credits-animated figure on slide 1 created by:: Preston Nichols, Wittenberg University. “Antimated Conic Section." [Online image] 29 December 2004. <http://http://www4.wittenberg.edu/academics/mathcomp/nicholswww/mmamisc/mmaexport.html>.