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
• .
A conic section (or simply
conic) is a curve obtained
as the intersection of the
surface of a cone with a
plane.
A cone has two identically
shaped parts called
nappes.

6. A LITTLE
HISTORY
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

7. TYPES OF
CONIC SECTIONS
There are mainly four types
of conic sections :
1.Parabola
2.Ellipse
3.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
i. 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.
ii. Parabolic mirrors are used to converge light
beams at the focus of the parabola.
Parabolic microphones perform a similar
function with sound waves.
iii. Solar ovens use parabolic mirrors to converge
light beams to use for heating.
iv. Parabola was used back in the medeival period
to navigate the path of a canon ball to attack
the enemy.
v. Parabola is the mathematical curve used by

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 PL
ANE

14. APPLICATIONS OF
ELLIPSES
i. 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.
ii. In bicycles, elliptical chains may be used for
mechanical advantage.
iii. 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.
iv. 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

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. APPLICATIO
NS OF
CIRCLES
i. Circles are used as wheels on cars, bikes and other
forms of transportation. Th eshape of a circle helps
create a smooth motion.
ii. Ferris wheels are circular.
iii. Gears and CDs which were, in their time, essential to
every day life are circles.
iv. 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. HYPERBOL
A
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
i. Hyperbolas are used in a navigation
system known as LORAN (long
range navigation)
ii. Hyperbolic as well as parabolic
mirrors and lenses are used in
systems of telescopes
iii.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 2b2/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.

23. BIBLIOGRAPHY
1) https://courses.lumenlearning.com
2) https://www.mathsisfun.com
3) http://mathcentral.uregina.ca
4) ISC Mathematics by S Chand Scool Books
5) http://xahlee.info
6) https://www.pleacher.com
7) https://www.slideshare.net

24. THANK YOU