Ellipse
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This document discusses analytic geometry and ellipses. It begins with objectives of defining key terms of conic sections like ellipses, and solving equations and real-world problems involving ellipses. An activity is described where students can draw an ellipse using a rubber band stretched between two pins. Examples of completed ellipses are shown along with their equations and key properties labeled like foci, vertices, axes, and eccentricity. Students are given practice problems to find properties of ellipses based on given information.
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Ellipse
- 1. ANALYTIC GEOMETRY James Bernhardt P. Perpetua
- 2. OBJECTIVES: Define the key terms in the Conic section such as ellipse, eccentricity, minor and major axes, foci, vertices, directrices and center through the discussion and activities provided; Solve equations and real world representations involving ellipse through the learning activities; Appreciate the importance of the Conic Section: Ellipse in real-world situations and applications through collaboration.
- 3. ACTIVITY 1 Materials: (a) Rubberband (b) 2 tacking pins (c) ball pen (d) piece of paper (e) Chipboard. Instruction: ■ Lay the paper on a flat chipboard. ■ Place the rubberband into a tacking pin. ■ Tack the pins on the paper, keeping the pins a desired distance apart. ■ Insert the ball pen point so that the string is keep taut. ■ Allow the pen to move along the inside of the string around the pins until you get back to the starting position keeping the string taut throughout.
- 4. Move one pin farther from the other pin, and sketch another one. What is effect on the curve when there is more separation between the two tacking pins?
- 5. ELLIPSE
- 7. 𝑥2 25 + 𝑦2 9 = 1 𝑎2 = 25 𝑏2 = 9 𝑎 = 5 𝑏 = 3 (0, −5) (0,5) (0, −3) (0,3) 𝐹𝑜𝑐𝑖 𝑐2 = 𝑎2 − 𝑏2 𝑐2 = 25 − 9 𝑐2 = 16 𝑐 = 4 (0, −4) (0,4)
- 8. Terms: Major axis –The line that pass through the foci. Minor axis –The perpendicular bisector of the major axis between the foci. Foci – Fixed points in the ellipse. Vertices –The points at which an ellipse makes its sharpest turns. Directrices –The ratio of distance of each point from a focus to the distance of that point from a fixed line is always the same. Center – A point inside the ellipse which is the midpoint of the line segment linking the two foci. 𝑎 𝑎 𝑐 𝑐𝑏 𝑏
- 9. Eccentricity -The distance between its center and either of its two foci. 𝑒 = 𝑐 𝑎
- 10. 𝑥2 64 + 𝑦2 25 = 1 𝑎2 = 64 𝑏2 = 25 𝑎 = 8 𝑏 = 5 (0, −5) (0,5) (√39,0) (√39,0)(−8,0) (8,0) 𝐹𝑜𝑐𝑖 𝑐2 = 𝑎2 − 𝑏2 𝑐2 = 64 − 25 𝑐2 = 39 𝑐 = 39
- 11. Quiz #1 Based on the prior activity, place what you have sketched inside the Cartesian plane. Instructions: 1. Make a Cartesian plane on the same paper where you have sketched using 1x1 cm as the distance from one point to another. (e.g) 2. Find the center of your ellipse. Place it on the point of origin. 3. Find the coordinates of the foci, vertices, co-vertices and their distances from the center. (e.g. 𝑎2 =? , 𝑏2 =? 𝑎𝑛𝑑 𝑐2 =? .)
- 12. (𝑥 − 3)2 +(𝑦 − 2)2 = 1(𝑥 − 3)2 9 + (𝑦 − 2)2 25 = 1
- 13. (𝑥 − 3)2 9 + (𝑦 − 2)2 25 = 1 𝑎2 = 25 𝑏2 = 9 𝑎 = 5 𝑏 = 3 (3,2) (6,2)(0,2) (3,7) (3, −3) 𝑐2 = 𝑎2 − 𝑏2 𝑐2 = 25 − 9 𝑐2 = 16 𝑐 = 4 (3,6) (3, −2)
- 14. (𝑥 − 2)2 9 + 𝑦2 16 = 1 (2,0) 𝑎2 = 16 𝑏2 = 9 𝑎 = 4 𝑏 = 3 (5,0)(−1,0) (2,4) (2, −4) 𝑐2 = 𝑎2 − 𝑏2 𝑐2 = 16 − 9 𝑐2 = 7 𝑐 = ± 7 (2, − 7) (2, 7)
- 15. Equation Major Axis Minor Axis Center Foci Vertices Directices Axis of Symmetry 𝑥2 𝑎² + 𝑦² 𝑏² = 1 x-axis y-axis (0,0) (±c,0) (±a,0) 𝑥 = 𝑎 𝑒 𝑎𝑛𝑑 𝑥 = −𝑎 𝑒 both x and y axes 𝑥2 𝑏² + 𝑦² 𝑎² = 1 y-axis x-axis (0,0) (0,±c) (0,±a) 𝑦 = 𝑎 𝑒 𝑎𝑛𝑑 𝑦 = −𝑎 𝑒 both x and y axes Equation Foci Vertices Co- Vertices Directices Axis of Symmetry (𝑥 − ℎ)2 𝑎² + (𝑦 − 𝑘)² 𝑏² = 1 (h±c,k) (h±a,k) (h,k±b) 𝑥 = h + 𝑎 𝑒 𝑎𝑛𝑑 𝑥 = ℎ − 𝑎 𝑒 both x and y axes (𝑥 − ℎ)2 𝑏² + (𝑦 − 𝑘)² 𝑎² = 1 (h,k±c) (h,k±a) (h±b,k) 𝑦 = 𝑘 + 𝑎 𝑒 𝑎𝑛𝑑 𝑦 = k − 𝑎 𝑒 both x and y axes Center at origin (0,0) Center at (h,k)
- 16. Instruction. Find the major axis, minor axis, eccentricity, the coordinates of the foci and vertices, length of the major and minor axes, and the directrices. 1. 16𝑥2 + 9𝑦2 = 144 2. 16𝑥2 + 25𝑦2 = 400 3. 9𝑥2 + 𝑦2 = 9 4. 3𝑥2 + 5𝑦2 = 15 5. 𝑥2 + 4𝑦2 = 4
- 17. Instruction. Find the equation of the following. 1. foci (-1,4), (-1,6) and vertex (-1,8) 2. foci (-2,3), (-2,9) and vertex (-2,12) 3. foci (3,-1), (8,-1) and vertex (10, -1) 4. foci (4,0), (-4,0) and vertex (5,0) 5. foci (0,8), (0,-8), eccentricity 4/5
Editor's Notes
- The set of all points, the sum of whose distances to two fixed points, called the foci, is constant.