Скачать или смотреть The Concept of Eccentricity (Precalculus / Conic Sections / Lesson 8)

In this lesson, we learn two definitions of the eccentricity of a conic (and in particular, an ellipse):
1. Eccentricity is the ratio (distance from point to focus)/(distance from point to directrix).
2. Eccentricity is the ratio (distance from center to focus) / (distance from center to endpoint of major axis)

We show why these are equivalent definitions, and get intuition and practice.

0:00 Introduction and two earlier perspectives on ellipses
0:40 A new perspective on ellipses (focus/directrix)
2:10 Task 1 (Writing an equation for an ellipse using the focus/directrix)
2:47 Check 1
3:34 Graphing the result from Check 1
5:08 Two definitions of eccentricity
7:07 Proving the two definitions are equal
8:02 Task 2 (Ellipse review / Using the c/a or c/b eccentricity definition)
9:08 Check 2
11:18 Summary
11:59 Appendix 1 (sqrt(x^2) ≠ x when x is negative)
12:50 Appendix 2 (The missing algebraic steps in the proof that the two eccentricity definitions were equivalent)