2. • Appolonious of Perga, a 3rd century B.C.
Greek geometer, wrote the greatest treatise on
the curves, his work "Conics" was the first to
show how all three curves, along with the
circle, could be obtained by slicing the same
right circular cone at continuously varying
angles
• Conics was discovered by Menaechmus, tutor
to Alexander the Great, over 2000 years ago
APPOLONIUS APPOLONIUS’ OPTICS

3. •Euclid’s works, “Optics” and “Elements” also
indicated knowledge of conics
EUCLID’S OPTICS
•Descarte and Fermat's co-ordinate geometry
included the many concepts introduced by the
ancient greeks
•Kepler’s laws of planetary motion brought conics
into new light, from being taken for granted, to
being marveled at

4. The conic sections are the non-
degenerate curves generated by the
intersections of a plane with one or
two napes of a cone.
For a plane perpendicular to the axis
of the cone, a circle is produced
For a plane which is not
perpendicular to the axis and which
intersects
only a single nape, the curve
produced is either an ellipse or a
parabola
The curve produced by a plane
intersecting both napes is a
hyperbola
The ellipse and hyperbola are know

5. Given a line D and a point F not on D,
conics is the locus of points P such that:
the distance from P to F divided by the distance from P
to D is a constant. That is,
distance[P,F]/distance[P,D] == e .
F is called the focus of the conic, D the directrix, and e
the eccentricity .
If 0 < e < 1, the conics is an ellipse
if e=0,it is a circle
If e == 1, it is a parabola
If e > 1, it is a hyperbola

6. A circle is a symmetrical figure, which can be formed
when a plane cuts a right circular cone parallel to its
base.
A circle is actually a special case of an ellipse, with
eccentricity = 0, that is, as if the two foci of the
ellipse are one.
For a circle at centre (x1,y1) and radius r, we can
give the equation as:
(x – x1) 2 + (y – y1) 2 = r 2
EQUATION (FROM x2+y2=r2
CENTRE)
ECCENTRICITY 0
RELATION TO FOCUS P=0

7. An ellipse is formed when a plane cuts a cone at an angle
The eccentricity e of an ellipse is defined as e := c/a, where c
is half the distance between foci.
For any ellipse, 0 < e < 1.
Any cylinder sliced on an angle will reveal an ellipse in cross-
section (as seen in the Tycho Brahe Planetarium in
Copenhagen).
For an ellipse with centre at the origin, with c and d as any
constants, the equation is: x2 + y2 = 1
c2 d2

8. In the 17th century, Kepler
discovered that planets moved about
the sun in elliptical orbits, with the
sun at one of their foci.
The orbits of the moon and of
artificial satellites of the earth are
also elliptical as are the paths of
comets in permanent orbit around the
sun
Halley’s comet takes about 76 years
to orbit the sun in its elliptical orbit.
This was correctly predicted by
Edmund Halley.
The electrons of the atom travel in
approximately elliptical orbits with the
nucleus as a focus.

9. Any light or signal that starts at one focus of
an ellipse will be reflected to the other focus.
This principle is used in lithotripsy , a medical
procedure for treating kidney stones.
The principle is also used in the construction
of "whispering galleries" where if a person
whispers near one focus, he can be heard at
the other focus, although he cannot be heard
at many places in between.
Equation (from centre) x2 + y2 = 1
c2 d2
Eccentricity 0<e<1
Relation to focus a2 - b2 = c2

10. •Pool shots are easier on an
elliptical table.
•The surface of the liquid in
a cylindrical glass will
appear elliptical when it is
tilted at an angle.

11. If the plane cuts a cone so that it lies at the same
angle as the slope of the cone then the intersection
is a parabola
A common definition defines it as the locus of
points P such that the distance from a line (called
the directrix) to P is equal to the distance from P to
a fixed point F (called the focus).
Parabola has eccentricity e:=1
A segment of a parabola is called a Lissajous
curve.
Equation (from 4px = y 2
centre)
Eccentricity
c/a
= 1
Relation to
focus p =
p

12. A body falling under the pull of gravity is a very
common approximation of a parabolic path
In the case of water from a water fountain,
every molecule of water follows a parabolic path
This discovery by Galileo in the 17th century
made it possible for cannoneers to work out the
kind of path a cannonball would travel if it were
hurtled through the air at a specific angle
This knowledge is used today in military,
aeronautics and sports to improve performance

13. •Parabolic reflectors have interesting properties. They can transform
any light from the focus to a straight beam.
• They are thus used in car lights.
Satellite dishes and antennae work on the reverse principal.
They can focus parallel rays of light on a focus to provide high
concentration.
Solar Furnaces use parabolic reflectors which focus sunlight such
that the temperature reaches very high extremes.

14. A plane which cuts a double ended code
through both its napes projects a
hyperbola
Hyperbola is commonly defined as the
locus of points P such that the difference
of the distances from P to two fixed
points F1, F2 (called foci) are constant.
That is, Abs[ distance[P,F1] -
distance[P,F2] ] == 2 a,
where a is a constant.
The eccentricity of a hyperbola is c/a > 1
(where c is the distance between the foci)
Equation x2 - y2 = 1
(from centre)
c2
d2
Eccentricity e>1
Relation to a2 - b2 = c2
focus

15. •A hyperbola can be observed in physical
situations like the light projected by a lamp,
or while sharpening a pencil
•A sonic boom shock wave has the shape of
a cone, and it intersects the ground in part of
a hyperbola.
•A hyperbola revolving around its axis forms
a surface called a hyperboloid. Certain
Buildings like nuclear reactors are
constructed as hyperboloids.

16. PRESENTED BY:
PRITIKA NILARATNA
GEETIKA AGARWAL
PRERIT JAIN
MENTORED BY:
HIMANI ASIJA
(PGT MATHEMATICS)
DELHI PUBLIC SCHOOL
VASANT KUNJ