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1. CONIC SECTIONS

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

3. TYPES OF CONIC SECTIONS

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)

5. (h,k)
C
P(x,y)
O (0,0)
x² + y² = r² (x – h) ² + (y – k) ² = r²
α β

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.

9. TYPES OF CONIC SECTIONS

10. ELLIPSE
AN ELLIPSE IS THE
SET OF ALL THE
POINTS ON A PLANE,
WHOSE SUM OF
DISTANCES FROM
TWO FIXED TWO
REMAINS CONSTACT.
P
P P
F F
¹
³²
²¹

11. α β
O
(0,c)
(0,-c)
(-b,0) (b,0)
(0,-a)
(0,a)
x² y²
a² b²
— —+ = 1
—+
x² y²
b² a²
— = 1
(-c ,0) (c, 0)

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. .

15. TYPES OF CONIC SECTIONS

16. 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 4O
P1
P2

17. α
β
F(a,0)O
x=-a
y² = 4ax
X' X
Y'
Y
F(-a,0) O
x=+a
y² = -4ax
X' X
Y'
Y
F(0,-a)
O
y = a
x² = 4ay
X' X
Y'
Y
F(0,a)
O
y = -a
x² = -4ay
X' X
Y'
Y

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.
 Automobile 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. Golden Gate Bridge in San
Francisco.
The suspension cable is a
parabola.
A concrete arch bridge.

21. TYPES OF CONIC SECTIONS

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

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
When 0 ≤ β < α; the plane cuts through both the nappes
& the curves of intersection is a hyperbola
x² y²
a² b²
— —- = 1
-
y² x²
a² b²
— —- = 1

25. 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.
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.

26. HYPERBOLIC PARABOLOID

27. THANK YOU
Efforts By :
Divyanshu Tyagi
XI - C