Ellipse three Examples and Solutions.pptx

This PowerPoint presentation is specifically designed to showcase and explain various examples of ellipses, both mathematical and real-life. It aims to help students recognize how ellipses appear not only in equations and coordinate graphs but also in the world around them. The presentation includes illustrative examples, solved problems, and visual representations that demonstrate the different forms and properties of ellipses.

Students will explore how ellipses are represented algebraically through standard and general equations, and how their key components—such as the center, foci, vertices, and axes—relate to these equations. Each example is presented with step-by-step solutions to guide learners in identifying important features and applying relevant formulas accurately.

Beyond mathematical examples, the presentation highlights real-world applications of ellipses, such as the paths of planets around the sun, satellite orbits, whispering galleries, reflective properties in architecture, and design elements in art and engineering. These examples aim to deepen students’ appreciation for the role of ellipses in science, technology, and everyday life.

Through visual aids, guided practice, and conceptual discussions, this PowerPoint presentation serves as a comprehensive learning tool that enhances students’ understanding and engagement with the topic of ellipses.

1. Ellipse Examples

2. Ellipse: Table of Formulas  

3. Example 1 Find a, b, and c: a² = 49 → a = 7 b² = 9 → b = 3 c² = a² - b² = 49 - 9 = 40 → c = √40 ≈ 6.3 2  

4. Example 1 Orientation: Horizontal Center: (3, 5) Vertices: V 1 (3 + 7 , 5 ) = ( 10 , 5 ), V 2 (3 -7 , 5) = ( -4 , 5 ) Co-vertices: W 1 (3, 5 +3 ) = ( 3 , 8 ), W 2 (3, 5 -3 ) = ( 3 , 2 ) Foci: F 1 (3 +6.3 2 , 5) = ( 9.32 , 5 ), F 2 (3 -6.3 2 , 5) = ( - 3 .32 , 5 )  

5. Example 2  

6. Example 2 Find a, b, and c: a² = 9 → a = 3 b² = 4 → b = 2 c² = a² - b² = 9 - 4 = 5 → c = √5 ≈ 2.2 4  

7. Example 2 Orientation: Horizontal Center: (4, 3) Vertices: V 1 (4+3, 3) = (7, 3), V 2 (4-3, 3) = (1, 3) Co-vertices: W 1 (4, 3+2) = (4, 5), W 2 (4, 3-2) = (4, 1) Foci: F 1 (4+ 2.24 , 3) ≈ (6.24, 3), F 2 (4- 2.24 , 3) ≈ (1.76, 3)