Conic Sections (Class 11 Project)

Types of Conic sections ; applications ; eccentricity ; ISC

1. Mathematics project Submitted to – kamal soni sir Submitted by – kushagra agrawal

2. acknowledgement   The success and final outcome of this project required a lot of guidance and assistance from many people and I am extremely privileged to have got this all along the completion of my project. All that I have done is only due to such supervision and assistance and I would not forget to thank them.   I respect and thank Kamal Soni Sir , for providing me an opportunity to do the project work in The Sanskaar Valley School and giving us all support and guidance which made me complete the project duly. I am extremely thankful to him for providing such a nice support and guidance, although he had busy schedule.

3. Index No. Title  Slide No. 1. Question 4 2. Conic Sections 5 3. Parabola  9 4. Ellipse  12 5. Hyperbola  17 6. Latus Rectum & Eccentricity 20 7. Bibliography  23

4. Question Construct different types of conics by PowerPoint presentation using the concept of double cone and a plane.

5. Conic Sections .

6. A little history The discovery of the conic sections is attributed to the Greek mathematician Manaechmus (circa 375-N325 B.C.) who was a tutor to Alexander the Great.  The conics were first defined as the intersection of: a right circular cone of varying vertex angle; a plane perpendicular to a element of the cone. (An element of a cone is any line that makes up the cone) Depending the angle is less than, equal to, or greater than 90 degrees, we get ellipse, parabola, or hyperbola respectively.  They were conceived in a attempt to solve the three famous problems of trisecting the angle, duplicating the cube, and squaring the circle. The definition of conic sections which we shall use is attributed to Apollonius. He is also the one to give the name ellipse, parabola, and hyperbola. In Renaissance, Kepler's law of planetary motion, Descarte and Fermat's coordinate geometry, and the beginning of projective geometry started by Desargues, La Hire, Pascal pushed conics to a high level. Many later mathematicians have also made contribution to conics, espcially in the development of projective geometry where conics are fundamental objects as circles in Greek geometry. Among the contributors, we may find Newton, Dandelin, Gergonne, Poncelet, Brianchon, Dupin, Chasles, and Steiner.

7. Types of conic sections There are mainly four types  of conic sections :  Parabola  Ellipse  Hyperbola 

8. Common parts of conic sections While each type of conic section looks very different, they have some features in common. For example, each type has at least one focus and directrix . Focus : It is a fixed point about which the conic section is constructed. It is a point about which rays reflected from the curve converge. A parabola has one focus; an ellipse and hyperbola have two. Directrix :  A directrix is a fixed straight line used to construct and define a conic section. As with the focus, a parabola has one directrix, while ellipses and hyperbolas have two.  

9. Parabola A parabola is formed when the plane is parallel to the surface of the cone, resulting in a U-shaped curve that lies on the plane.  It is the set of all points whose distance from the focus, is  equal  to the distance from the directrix. The point halfway between the focus and the directrix is called the vertex of the parabola. Every parabola has certain features: An axis of symmetry, which is a line connecting the vertex and the focus which divides the parabola into two equal halves All parabolas possess an eccentricity value e=1.

10. four parabolas are graphed as they appear on the coordinate plane. They may open up, down, to the left, or to the right.

11. Applications of parabola Prabolas is the path of any object thrown in the air. Parabola is formed when a football is kicked, a baseball is hit, a basketball hoop is made, dolphins jump, etc. Parabolic mirrors are used to converge light beams at the focus of the parabola. Parabolic microphones perform a similar function with sound waves. Solar ovens use parabolic mirrors to converge light beams to use for heating.  Parabola was used back in the medeival period to navigate the path of a canon ball to attack the enemy. Parabola is the mathematical curve used by engineers in designing some suspension bridges.

12. ellipse    When the plane’s angle relative to the cone is between the outside surface of the cone and the base of the cone, the resulting intersection is an ellipse.    It is the set of all points for which the sum of the distances from two fixed points (the foci) is constant. In the case of an ellipse, there are two foci, and two directrices. Ellipses have these features: A major axis, which is the longest width across the ellipse A minor axis, which is the shortest width across the ellipse A center, which is the intersection of the two axes Two focal points —for any point on the ellipse, the sum of the distances to both focal points is a constant

13. a typical ellipse is graphed below as it appears on the coordinate plane

14. Applications of ellipses They are used in astronomy to describe the shapes of the orbits of objects in space. All the planets in our solar system revolves around the sun in elliptical orbits.  In bicycles, elliptical chains may be used for mechanical advantage.  Elliptical Pool Table : The reflection property of the ellipse is useful in elliptical pool — if you hit the ball so that it goes through one focus, it will reflect off the ellipse and go into the hole which is located at the other focus. Whispering galleries : these galleries have circular or elliptical ceilings which amplifies the faintest whisperes so they can be heard in all parts of the gallery. Examples of such galleries are Grand Central Terminal, St paul's Cathedral and Gol Gumbaz.

15. circle A circle is formed when the plane is parallel to the base of the cone. Its intersection with the cone is therefore a set of points equidistant from a common point. The definition of an ellipse includes being parallel to the base of the cone as well, so all circles are a special case of the ellipse.  All circles have certain features: A center point A radius, which the distance from any point on the circle to the center point

16. Applications of circles Circles are used as wheels on cars, bikes and other forms of transportation. Th eshape of a circle helps create a smooth motion. Ferris wheels are circular. Gears and CDs which were, in their time, essential to every day life are circles. Circles have largest possible ratio of area to perimeter and therefore are used in a variety of things including bottles, pipelines, etc as they would require lesser material as compared to any other shape. 

17. Hyperbola A hyperbola is formed when the plane is parallel to the cone’s central axis, meaning it intersects both parts of the double cone. A hyperbola is the set of all points where the difference between their distances from two fixed points (the foci) is constant. In the case of a hyperbola, there are two foci and two directrices. Hyperbolas have these features: A center, which bisects every chord that passes through it. Two focal points, around which each of the two branches bend Two vertices, one for each branch

18. A graph of a typical hyperbola

19. Applications of hyperbola Hyperbolas are used in a navigation system known as LORAN (long range navigation) Hyperbolic as well as parabolic mirrors and lenses are used in systems of telescopes The hyperboloid is the design standard for all nuclear cooling towers and some coal-fired power plants. It allows the structutre to withstand high winds and can be built with relatively lesser material.

20. Latus rectum The latus rectum is the chord through the focus, and parallel to the directrix. The length of Latus Rectum in a  Parabola, is four times the focal length Circle, is the diameter Ellipse, is 2b 2 /a (where a and b are one half of the major and minor diameter).

21. eccentricity The eccentricity, denoted ee, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular. The eccentricity of a conic section is defined to be the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the nearest directrix. The value of e is constant for any conic section. The value of e can be used to determine the type of conic section as well: if e=1, the conic is a parabola If e<1, it is an ellipse If e=0, it is a circle If e>1, it is a hyperbola Note that two conic sections are similar (identically shaped) if and only if they have the same eccentricity.

22. . Hyperbolas and non-circular ellipses have two foci and two associated directrices, while parabolas have one focus and one directrix. In the figure, each type of conic section is graphed with a focus and directrix. The orange lines denote the distance between the focus and points on the conic section, as well as the distance between the same points and the directrix. These are the distances used to find the eccentricity.

23. Bibliography https://courses.lumenlearning.com https://www.mathsisfun.com http://mathcentral.uregina.ca ISC Mathematics by S Chand Scool Books http://xahlee.info https://www.pleacher.com   https://www.slideshare.net  

24. Thank you