Conic Sections- Circle, Parabola, Ellipse, Hyperbola

Conic Sections- Circle, Parabola, Ellipse, Hyperbola all topics covered. this is a presentation with excelent animations and explatition.

1. CONIC SECTIONS XI C

2. α β THE INTERSECTION OF A PLANE WITH A CONE, THE SECTION SO OBTAINED IS CALLED A CONIC SECTION V m Lower nappe Upper nappe Axis Generator l This is a conic section.

3. TYPES OF CONIC SECTIONS CIRCLE ELLIPSE PARABOLA HYPERBOLA

4. CIRCLE A CIRCLE IS THE SET OF ALL POINTS ON A PLANE THAT ARE EQUIDISTANT FROM A FIXED POINT ON A PLANE. O P (x,y) Radius

5. (h,k) C P (x,y) O (0,0) x² + y ² = r ² (x – h) ² + (y – k) ² = r ² α β When β = 90°, the section is a circle Standard Equation General Equation

6. APPLICATIONS

7.   One prime example of a circle that you can find in real life is a Ferris Wheel.  All the points along the outer rim of the wheel are equidistant from the center.  The lights on this one can help you see that a little easier.  Another good example of circles are bicycle wheels.  Circles are the best shape for a bicycle because they roll very easily because they are round.  The center point would be the (h,k) in the equation and all the points along the outer edge could be the (x,y) values.  The radius could be represented by the bars supporting the wheel that run from the center to the outer rim.  

8. FERRIS WHEEL BICYCLE WHEELS

9. TYPES OF CONIC SECTIONS CIRCLE ELLIPSE PARABOLA HYPERBOLA

10. ELLIPSE AN ELLIPSE IS THE SET OF ALL THE POINTS ON A PLANE, WHOSE SUM OF DISTANCES FROM TWO FIXED TWO REMAINS CONSTANT. P P P F F ¹ ³ ² ² ¹

11. α β O (0,c) (0,-c) (-b,0) (b,0) (0,-a) (0,a) x² y² a ² b ² — — + = 1 — + x² y² b ² a² — = 1 (-c ,0) (c, 0) When α < β < 90°, the section is an ellipse Vertical Ellipse Horizontal Ellipse (0,-b) (0,b) ( a ,0) (-a,0)

12. APPLICATIONS

13. Elliptical forms have many applications: orbits of satellites, planets, and comets shapes of galaxies; gears and cams, some airplane wings, boat keels, and rudder; tabletops; public fountains; and domes in buildings are a few example. In the 17th century, Johannes Kepler discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun at one focus, in his first law of planetary motion. Later, Isaac Newton explained this as a corollary of his law of universal gravitation . Keplerian elliptical orbits are the result of any radially-directed attraction force whose strength is inversely proportional to the square of the distance. Thus, in principle, the motion of two oppositely-charged particles in empty space would also be an ellipse.

14. A pair of elliptical gears with pivot points at foci . A n elliptical dome . A pplication in medicine is the use of elliptical reflectors and ultrasound to break up kidney stones. Elliptical orbit of Earth around Sun

15. TYPES OF CONIC SECTIONS CIRCLE ELLIPSE PARABOLA HYPERBOLA

16. PARABOLA A PARABOLA IS THE SET OF ALL POINTS IN A PLANE THAT ARE EQUIDISTANT FROM A FIXED POINT A B V PARABOLA ( VERTEX ) F ( focus ) 1 2 3 4 O P 1 P 2

17. α β F (a,0) O x = -a y ² = 4ax X' X Y' Y F (-a,0) O x = +a y ² = -4ax X' X Y ' Y F (0,-a) O y = a x ² = 4ay X' X Y ' Y F (0,a) O y = -a x ² = -4ay X' X Y ' Y When α = β, the section is an parabola Horizontal Parabola Horizontal Parabola Vertical Parabola Vertical Parabola

18. APPLICATIONS

19. Parabolic reflector used in all reflecting telescopes from 3- to 6-inch . Pome types to the 200-inch research instrument on Mount Palomar in California . Parallel light rays from distant celestial bodies are reflected to the focus off a parabolic mirror . If the light source is the sun, then the parallel rays are focused at F and we have a solar furnace. A utomobile headlights can use parabolic reflectors with special lenses over the light to diffuse the rays into useful patterns. Parabolic forms are frequently encountered in the physical world. Suspension bridges. arch bridges, microphones, symphony shells, satellite antennas, radio and optical telescopes, radar equipment, solar furnaces, and searchlights are only a few of many items that use parabolic forms in their design.

20. A concrete arch bridge. Solar Reflectors Golden Gate Bridge in SanFrancisco . The suspension cable is a parabola.

21. TYPES OF CONIC SECTIONS CIRCLE ELLIPSE PARABOLA HYPERBOLA

22. HYPERBOLA F ( focus ) V (vertex) A B A HYPERBOLA IS THE SET OF ALL POINTS,THE DIFFERENCE OF WHOSE DISTANCES FROM TWO FIXED POINTS IS CONSTANT V (vertex) F ( focus )

23. α β Transverse axis F Conjugate axis F (c ,0) (a ,0) ( -c ,0) (-a ,0) O F F (0 ,c) (0 ,a) (0 ,-c) (0 ,-a) O ¹ ¹ ² ² x² y² a ² b ² — — - = 1 - y² x² a ² b ² — — - = 1 When 0 ≤ β < α; the plane cuts through both the nappes & the curves of intersection is a hyperbola