1. 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 (

2. 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

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

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

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

6. Example
Distance Formula:

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

8. 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

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

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

11. 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

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

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

14. Try This
Complete the square for:
then factor your result

15. Example
Complete the square for:
then factor your result

16. 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:

17. Try This
Solve by completing the square:

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

19. Try This
Solve by completing the square…

20. 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 <