“Conic section” is a fundamental of the Mathematics. This report is made from my studying about the conic section in the Mathematics books and on the internet. This report contains topics that involve with conic section such as: The history of Conic section studying, Parabola, Ellipse, Hyperbola and their applications with figures may help you to understand easily. This report is may use to refer for next time and its can be usefulness for the readers.

1. M a t h e m a t i c s r e p o r t
CCoonniicc SSeeccttiioonn
Submitted by
Mr. Teekayu Ruannark
Number 7 Secondary 6/6
Supervised by : Mrs. Vannee Chatngern
This report is the part of Mathematics (M43102) 2nd
Semester Academic Year: 2009
Benchamarachuthit Chanthaburi School Chanthaburi Education Service Area Office 1

2. PPrreeffaaccee
“Conic section” is a fundamental of the Mathematics. This
report is made from my studying about the conic section in the
Mathematics books and on the internet. This report contains
topics that involve with conic section such as: The history of Conic
section studying, Parabola, Ellipse, Hyperbola and their
applications with figures may help you to understand easily.
This report is may use to refer for next time and its can be
usefulness for the readers.
Thank you
Mr. Teekayu Ruannark
writer

3. CCoonntteennttss
Page
Conic section 1
History 1
Parabola 5
Analysis 5
Equations 6
Applications 8
Ellipse 10
Analysis 10
Equations 11
Applications 15
Hyperbola 18
Equations 19
Applications 21
References 24

4. 1
Apollonius of Perga [Pergaeus]
(Ancient Greek: Ἀπολλώνιος)
(ca. 262 BC–ca. 190 BC)
was a Greek geometer and astronomer noted
for his writings on conic sections. His innovative
methodology and terminology, especially in the
field of conics, influenced many later scholars
including Ptolemy, Francesco Maurolico, Isaac
Newton, and René Descartes. It was Apollonius
who gave the ellipse, the parabola, and the
hyperbola the names by which we know them.
The hypothesis of eccentric orbits, or
equivalently, deferent and epicycles, to explain
the apparent motion of the planets and the
varying speed of the Moon, are also attributed
to him. Apollonius' theorem demonstrates that
the two models are equivalent given the right
parameters. Ptolemy describes this theorem in
the Almagest XII.1. Apollonius also researched
the lunar theory, for which he is said to have
been called Epsilon (ε). The crater Apollonius on
the Moon is named in his honor.
CCoonniicc sseeccttiioonn
In mathematics, a conic section (or just conic) is a curve obtained by intersecting
a cone (more precisely, a right circular conical surface) with a plane. In analytic
geometry, a conic may be defined as a plane algebraic curve of degree 2. The conic
sections were named and studied as long ago as 200 BC, when Apollonius of Perga
undertook a systematic study of their
properties.
History
Menaechmus
It is believed that the first definition of a
conic section is due to Menaechmus. This work
does not survive, however, and is only known
through secondary accounts. The definition
used at that time differs from the one
commonly used today in that it requires the
plane cutting the cone to be perpendicular to
the line that generates the cone as a surface of
revolution. Thus the shape of the conic is
determined by the angle formed at the vertex
of the cone; If the angle is acute then the conic
is an ellipse, if the angle is right then the conic
is a parabola, and if the angle is obtuse then
the conic is a hyperbola. Note that the circle

5. 2
Euclid
(Greek: Εὐκλείδης — Eukleídēs)
, fl. 300 BC, also known as Euclid of Alexandria,
was a Greek mathematician and is often
referred to as the "Father of Geometry." He was
active in Hellenistic Alexandria during the reign
of Ptolemy I (323–283 BC). His Elements is the
most successful textbook and one of the most
influential works in the history of mathematics,
serving as the main textbook for teaching
mathematics (especially geometry) from the
time of its publication until the late 19th or early
20th century.[1][2][3] In it, the principles of
what is now called Euclidean geometry were
deduced from a small set of axioms. Euclid also
wrote works on perspective, conic sections,
spherical geometry, number theory and rigor.
"Euclid" is the anglicized version of the
Greek name Εὐκλείδης — Eukleídēs, meaning
"Good Glory".
cannot be defined this way and was not considered a conic at this time.
Euclid is said to have written four
books on conics but these were lost as well.
Archimedes is known to have studied conics
as well, having determined the area bounded
by a parabola and an ellipse. The only part of
this work to survive is a book on the solids of
revolution of conics.
Apollonius of Perga
The greatest progress in the study of
conics by the ancient Greeks is due to
Apollonius of Perga, whose eight volume
Conic Sections summarized the existing
knowledge at the time and greatly extended it.
Apollonius's major innovation was to
characterize a conic using properties within
the plane and intrinsic to the curve; this
greatly simplified analysis. With this tool, it
was now possible to show that any plane
cutting the cone, regardless of its angle, will
produce a conic according to the earlier
definition, leading to the definition commonly
used today.
Pappus is credited with discovering importance of the concept of a focus of a
conic, and the discovery of the related concept of a directrix.

6. 3
Omar Khayyam
Apollonius's work was translated into Arabic and much of his work only survives
through the Arabic version. Muslims found applications to the theory; the most notable
of these was the Persian mathematician and poet Omar Khayyam who used conic
sections to solve algebraic equations.
Europe
Johann Kepler extended the theory of conics through the "principle of
continuity", a precursor to the concept of limits. Girard Desargues and Blaise Pascal
developed a theory of conics using an early form of projective geometry and this help
provide impetus for the study of this new field. In particular, Pascal discovered a
theorem known as the hexagrammum mysticum from which many other properties of
conics can be deduced. Meanwhile, René Descartes applied his newly discovered
Analytic geometry to the study of conics. This had the effect of reducing the geometrical
problems of conics to problems in algebra.

7. 4

8. 5
Parabola
In mathematics, the parabola (pronounced
/pəˈræbələ/, from the Greek παραβολή) is a conic
section, the intersection of a right circular conical
surface and a plane parallel to a generating straight
line of that surface. Given a point (the focus) and a
line (the directrix) that lie in a plane, the locus of
points in that plane that are equidistant to them is a
parabola.
A particular case arises when the plane is
tangent to the conical surface of a circle. In this
case, the intersection is a degenerate parabola
consisting of a straight line.
The parabola is not an important concept in abstract mathematics, but it is also
seen with considerable frequency in the physical world, and there are many practical
applications for the construct in engineering, physics, and other domains.
Analysis
In Cartesian coordinates, a parabola with an axis parallel to the y axis with vertex
(h,k), focus (h,k + p), and directrix y = k - p, with p being the distance from the vertex to
the focus, has the equation
or, alternatively with axis parallel to the x-axis, focus (h + p,k), and directrix x = h − p,

9. 6
More generally, a parabola is a curve in the Cartesian plane defined by an irreducible
equation of the form
such that , where all of the coefficients are real, where or ,
and where more than one solution, defining a pair of points (x, y) on the parabola,
exists. That the equation is irreducible means it does not factor as a product of two not
necessarily distinct linear equations.
Equations
(with vertex (h, k) and distance p between vertex and focus - note that if the
vertex is below the focus, or equivalently above the directrix, p is positive, otherwise p is
negative; similarly with horizontal axis of symmetry p is positive if vertex is to the left of
the focus, or equivalently to the right of the directrix)

10. 7
Cartesian
Vertical axis of symmetry
.
Horizontal axis of symmetry
.
General parabola
The general form for a parabola is
This result is derived from the general conic equation (A'x2 + B'xy + C'y2 + D'x +
E'y + F' = 0) and the fact that, for a parabola, B'2 = 4A'C'.

11. 8
Applications
Parabolic forms are frequently encountered in the physical world. Suspension
bridges. arch bridges, microphones, symphony shells, satellite antennas, radio and
optical teleescopes, radar equipment, solar furnaces, and searchlights are only a few of
many items that use parabolic forms in their design.
Illustrates a parabolic reflector used in all reflecting telescopes—from 3- to 6-inch
home 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. Temperatures of over 6,000°C have been achieved by such furnaces. If
we locate a light source at F, then the rays in the figure reverse, and we have a
spotlight or a searchlight. Automobile headlights can use parabolic reflectors with
special lenses over the light to diffuse the rays into useful patterns.

12. 9
This figure shows a suspension bridge, such as the Golden Gate Bridge in San
Francisco. The suspension cable is a parabola. It is interesting to note that a free-
hanging cable, such as a telephone line, does not form a parabola. It forms another
curve called a catenary.
This figure shows a concrete arch bridge. If all the loads on the arch are to be
compression loads (concrete works very well under compression), then using physics
and advanced mathematics, it can be shown that the arch must be parabolic.

13. 10
Ellipse
In mathematics, an ellipse (from Greek
ἔλλειψις elleipsis, a "falling short") is the bounded
case of a conic section, the geometric shape that
results from cutting a circular conical or cylindrical
surface with an oblique plane (the two unbounded
cases being the parabola and the hyperbola). It is
also the locus of all points of the plane whose
distances to two fixed points (the foci) add to the
same constant.
Ellipses also arise as images of a circle or a sphere under parallel projection,
and some cases of perspective projection. Indeed, circles are special cases of ellipses.
An ellipse is also the closed and bounded case of an implicit curve of degree 2, and of
a rational curve of degree 2. It is also the simplest Lissajous figure, formed when the
horizontal and vertical motions are sinusoids with the same frequency.
Analysis
General ellipse
In analytic geometry, the ellipse is defined as the set of points (X,Y) of the
Cartesian plane that satisfy the implicit equation
provided that F is not zero and F(B2 − 4AC) is positive; or of the form
with

14. 11
Canonical form
By a proper choice of coordinate system, the ellipse can be described by the
canonical implicit equation
Here (x,y) are the point coordinates in the canonical system, whose origin is the
center (Xc,Yc) of the ellipse, whose x-axis is the unit vector (Xa,Ya) parallel to the major
axis, and whose y-axis is the perpendicular vector ( -Ya,Xa) That is, x = Xa(X -
Xc) + Ya(Y - Yc) and y = - Ya(X - Xc) + Xa(Y - Yc).
In this system, the center is the origin (0,0) and the foci are ( - ea,0) and ( + ea,0).
Any ellipse can be obtained by rotation and translation of a canonical ellipse with the
proper semi-diameters. Moreover, any canonical ellipse can be obtained by scaling the
unit circle of , defined by the equation
by factors a and b along the two axes.
For an ellipse in canonical form, we have
The distances from a point (X,Y) on the ellipse to the left and right foci are a + eX and
a - eX, respectively.
Equations
Ellipse is the set of all points in the plane, the sum of whose distances from two fixed
points, called the foci, is a constant. Foci Sometimes this definition is given in terms of
“a locus of points” or even “the locus of a point” satisfying this condition – it all means
the same thing.

15. 12
For reasons that will become apparent, we will denote the sum of these distances by
2a.
We see from the definition that an ellipse has two axes of symmetry, the larger of
which we call the major axis and the smaller the minor axis. The two points at the ends
of the ellipse (on the major axis) are called the vertices. It happens that the length of
the major axis is 2a, the sum of the distances from any point on the ellipse to its foci. If
we call the length of the minor axis 2b and the distance between the foci 2c, then the
Pythagorean Theorem yields the relationship b2 + c2 = a2:

16. 13
By imposing coordinate axes in this convenient manner, we see that the vertices are at
the x intercepts, at a and -a, and that the y-intercepts are at b and -b. Let the variable
point P on the ellipse be given the coordinates (x, y). We may then apply the distance
formula for the distances from P to F1 and from P to F2 to express our geometrical
definition of the ellipse in the language of algebra:
Substituting a2 – b2 for c2 and using a little algebra, we can then derive the standard
equation for an ellipse centered at the origin,
where a and b are the lengths of the semi-major and semi-minor axes, respectively. (If
the major axis of the ellipse is vertical, exchange a and b in the equation.) The points

17. 14
(a, 0) and (-a, 0) are called the vertices of the ellipse. If the ellipse is translated up/down
or left/right, so that its center is at (h, k), then the equation takes the form
If a = b, we have the special case of an ellipse whose foci coincide at the center – that
is, a circle of radius a.
The ellipse has the following remarkable reflection property. Let P be any point on
the ellipse, and construct the line segments joining P to the foci. Then these lines make
equal angles to the tangent line at P.

18. 15
Applications
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.
More generally, in the gravitational two-body problem, if the two bodies are
bound to each other (i.e., the total energy is negative), their orbits are similar ellipses
with the common barycenter being one of the foci of each ellipse. The other focus of
either ellipse has no known physical significance. Interestingly, the orbit of either body in
the reference frame of the other is also an ellipse, with the other body at one focus.
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.
(However, this conclusion ignores losses due to electromagnetic radiation and quantum
effects which become significant when the particles are moving at high speed.)
An elliptical orbit also results when the attraction force is inversely proportional to
distance from an attracting object. In this case the orbit differs from the Keplerian orbit
in that the object is placed at the centre of the ellipse.

19. 16
This figure shows a pair of elliptical gears with pivot points at foci. Such gears
transfer constant rotational speed to variable rotational speed, and vice versa.
This figure shows an elliptical dome. An interesting property of such a dome is
that a sound or light source at one focus will reflect off the dome and pass through
other focus. One of the chambers in the Capitol Building in Washington, D.C., has such
a dome, and is referred to as a whispering room because a whisperer one focus can be
easily heard at the other focus.

20. 17
A fairly recent application in medicine is the use of elliptical reflectors and
ultrasound to break up kidney stones. A device called a lithotripter is used togenerate
intense sound waves that break up the stone from outside the body, thus avoiding
surgery. To be certain that the waves do not damage other parts of the body, the
reflecting property of the ellipse is used to design and correctly position the lithotripter.

21. 18
Hyperbola
In mathematics a hyperbola is a
smooth planar curve having two connected
components or branches, each a mirror
image of the other and resembling two
infinite bows. The hyperbola is traditionally
described as one of the kinds of conic
section or intersection of a plane and a
cone, namely when the plane makes a
smaller angle with the axis of the cone than
does the cone itself , the other kinds being
the parabola and the ellipse (including the
circle).
Hyperbolas arise in practice in many
ways: as the curve representing the function
y = 1/x in the Cartesian plane, as the appearance of a circle viewed from within it, as
the path followed by the shadow of the tip of a sundial, as the shape of an open orbit
(as distinct from a closed and hence elliptical orbit) such as followed by a spacecraft
during a gravity assisted swing-by of a planet, more generally any spacecraft exceeding
the escape velocity of the nearest planet, or a single-apparition comet (one travelling
too fast to ever return to the solar system), or the scattering trajectory of a subatomic
particle (acted on by repulsive instead of attractive forces but the principle is the same),
and so on.
Each branch of the hyperbola consists of two arms which become straighter
(lower curvature) further out from the center of the hyperbola. Diagonally opposite arms
one from each branch tend in the limit to a common line, called the asymptote of those
two arms. There are therefore two asymptotes, whose intersection is at the center of

22. 19
symmetry of the hyperbola where it can be thought of as the mirror point about which
each branch reflects to form the other branch. In the case of the curve y = 1/x the
asymptotes are the two coordinate axes, while for the circle viewed from within they are
the tangents to the circle where the plane of the lens normal to the gaze cuts the circle:
without cuts the circle appears as an ellipse, or a parabola when the plane just grazes
the circle.
Hyperbolas share many of the ellipse's analytical properties such as eccentricity,
focus, and directrix. Typically the correspondence can be made with nothing more than
a change of sign in some term. Many other mathematical objects have their origin in the
hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids
("wastebaskets"), hyperbolic geometry (Lobachevsky's celebrated non-Euclidean
geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a non-
Euclidean geometry used in both relativity and quantum mechanics).
Equations
The hyperbola can be constructed by connecting the free end of a rigid bar
, where is a focus, and the other focus with a string . As the bar
is rotated about and is kept taut against the bar (i.e., lies on the bar), the locus
of is one branch of a hyperbola (left figure above; Wells 1991). A theorem of
Apollonius states that for a line segment tangent to the hyperbola at a point and
intersecting the asymptotes at points and , then is constant, and
(right figure above; Wells 1991).

23. 20
Let the point on the hyperbola have Cartesian coordinates , then the definition
of the hyperbola gives
Rearranging and completing the square gives
and dividing both sides by results in
By analogy with the definition of the ellipse, define
so the equation for a hyperbola with semimajor axis parallel to the x-axis and
semiminor axis parallel to the y-axis is given by

24. 21
or, for a center at the point instead of ,
Unlike the ellipse, no points of the hyperbola actually lie on the semiminor axis, but
rather the ratio determines the vertical scaling of the hyperbola. The eccentricity of
the hyperbola (which always satisfies ) is then defined as
In the standard equation of the hyperbola, the center is located at , the foci
are at , and the vertices are at . The so-called asymptotes
(shown as the dashed lines in the above figures) can be found by substituting 0 for the
1 on the right side of the general equation (8),
and therefore have slopes .
Applications
Hyperbolas may be seen in many sundials. On any given day, the sun revolves
in a circle on the celestial sphere, and its rays striking the point on a sundial traces out
a cone of light. The intersection of this cone with the horizontal plane of the ground
forms a conic section, by definition. At most populated latitudes and at most times of the
year, this conic section is a hyperbola. In practical terms, the shadow of the tip of a pole

25. 22
traces out a hyperbola on the ground over the course of a day. The shape of this
hyperbola varies with the geographical latitude and with the time of the year, since
those factors affect the cone of the sun's rays relative to the horizon. The collection of
such hyperbolas for a whole year at a given location was called a pelekinon by the
Greeks, since it resembles a double-bladed ax.
A hyperbola is the basis for solving trilateration problems, the task of locating a
point from the differences in its distances to given points — or, equivalently, the difference
in arrival times of synchronized signals between the point and the given points. Such
problems are important in navigation, particularly on water; a ship can locate its position
from the difference in arrival times of signals from a LORAN or GPS transmitters.
Conversely, a homing beacon or any transmitter can be located by comparing the
arrival times of its signals at two separate receiving stations; such techniques may be
used to track objects and people. In particular, the set of possible positions of a point
that has a distance difference of 2a from two given points is a hyperbola of vertex
separation 2a whose foci are the two given points.
The paths followed by any particle in the classical Kepler problem is a conic
section. In particular, if the total energy E of the particle is greater than zero (i.e., if the
particle is unbound), the path of such a particle is a hyperbola. This property is useful in
studying atomic and sub-atomic forces by scattering high-energy particles; for example,
the Rutherford experiment demonstrated the existence of an atomic nucleus by
examining the scattering of alpha particles from gold atoms. If the short-range nuclear
interactions are ignoreed, the atomic nucleus and the alpha particle interact only by a
repulsive Coulomb force, which satisfies the inverse square law requirement for a
Kepler problem.
As shown first by Apollonius of Perga, a hyperbola can be used to trisect any
angle, a intensely studied problem of geometry. Given an angle, one first draws a circle
centered on its middle point O, which intersects the legs of the angle at points A and B.

26. 23
One next draws the line through A and B and constructs a hyperbola of eccentricity ε=2
with that line as its transverse axis and B as one focus. The directrix of the hyperbola is
the bisector of AB, and for any point P on the hyperbola, the angle ABP is twice as
large as the angle BAP. Let P be a point on the circle. By the inscribed angle theorem,
the corresponding center angles are likewise related by a factor of two, AOP = 2×POB.
But AOP+POB equals the original angle AOB. Therefore, the angle has been trisected,
since 3×POB = AOB.

27. 24
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28. 25
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