5.03a - Ellipse at Origin | High School

University:
High School
Course:
High School Physics
Academic year:

2024
Views:

67

Pages:

18
Author:

1 1

5.03a The Ellipse Ellipses • An Ellipse is an elongated circle. Focus Points (Foci) F’ A circle has one focus, its centre. F The ellipse has two foci. Definitions Ellipse Centre Minor Axis Major Axis Definitions Ellipse The locus of a point P which moves such that the sum of the distances from the two fixed points (called the foci) is always constant. K = 𝑃1 𝐹 ′ + 𝑃1 𝐹 P1 F’ F P2 = 𝑃2 𝐹 ′ + 𝑃2 𝐹 = 𝑃3 𝐹 ′ + 𝑃3 𝐹 =⋯ Definitions Centre The Midpoint of the segment 𝐹 ′ 𝐹 𝑥 𝑦 F’ (-c,0) 𝑥1 + 𝑥2 𝑦1 + 𝑦2 𝑀𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝐹𝑜𝑟𝑚𝑢𝑙𝑎: , 2 2 F Centre (c,0) 𝑥=𝑦 =𝑐 Definitions Major Axis Segment on the transverse axis connecting two vertices Because this is the longest, it is the Major Axis We call half of this the Semi Major Axis A’ (-a,0) A (a,o) C Furthest possible locus points to center Definitions Minor Axis Segment on the conjugate axis connecting two vertices B (0,b) We call half of this the Semi Minor Axis C Closest possible locus points to center B’ (0,-b) Because this is the shortest, it is the Minor Axis Ellipses Centered at Origin Transverse is x-axis Transverse is y-axis Horizontal Ellipse Vertical Ellipse B(0,b) F’(-c,0) F(c,0) A’(-a,0) A (a,0) B’(0,-b) Focal Axis = 2a (along the x) Ellipses Centered at Origin Transverse is x-axis Transverse is y-axis Horizontal Ellipse Vertical Ellipse B(0,b) B(0,b) F’(-c,0) F(0,c) F(c,0) A’(-a,0) A (a,0) B’(0,-b) A’(-a,0) A (a,0) F’(0, -c) B’(0,-b) Focal Axis = 2a (along the x) Focal Axis = 2b (along the y) Ellipses Centered at Origin on x-axis Horizontal Ellipse B a (half sum of distance) b B(0,b) O F’(-c,0) F(c,0) A’(-a,0) A (a,0) B’(0,-b) Focal Axis = 2a (along the x) c F Ellipses Centered at Origin on y-axis F b c O Vertical Ellipse B(0,b) a F(0,c) A A’(-a,0) A (a,0) F’(0, -c) B’(0,-b) Focal Axis = 2b (along the y) Rules of Ellipses Centered at Origin on x-axis on y-axis Horizontal Ellipse Vertical Ellipse 2 2 𝑥 𝑦 + = 1 2 2 𝑎 𝑏 Example 1 𝑎>𝑏 3 4 Horizontal 𝑥2 𝑦2 + =1 2 2 (4) (3) 𝑥2 𝑦2 + =1 16 9 Example 1 3 4 Show in General Form - Common denominator 9𝑥 2 16𝑦 2 + =1 144 144 9𝑥 2 + 16𝑦 2 =1 144 𝑥2 𝑦2 + =1 16 9 9𝑥 2 + 16𝑦 2 = 144 9𝑥 2 + 16𝑦 2 − 144 = 0 Example 2 𝑎<𝑏 4 1 Vertical 𝑥2 𝑦2 + =1 2 2 (1) (4) 𝑥2 𝑦2 + =1 1 16 2 𝑦 𝑥2 + =1 16 Representing Inequalities 2 2 2 2 𝑥 𝑦 + < 1 𝑎2 𝑏2 𝑥 𝑦 + > 1 𝑎2 𝑏2 Shade inside and dotted Shade outside and dotted ≤1 Shade inside and solid ≥1 Shade outside and solid Example 3 2 3 𝑎>𝑏 HORIZONTAL 𝑥2 𝑦2 + =1 2 2 (3) (2) 𝑥2 𝑦2 + =1 9 4 𝑥2 𝑦2 + ≤1 9 4 Solid Line Shaded Inside 𝑥2 𝑦2 + ≤1 9 4 Homework • 5.03a – Ellipse Centered at the Origin • Visions 2 • P177: 3abde, 4ace, 6ace, 7bd, 9bc