CONIC SECTIONS
1- Circle
2- Ellipse
3- Parabola 4- Hyperbola
Done by :
- Mohamed Yousef AlQursi
- Hamad Ali Saleh
- Hassan Taher
- Mohamed Almansoori
12-7
Task 1 / Introduction
• The circle is one of the most common shapes in our daily life, and indeed
the universe. Planets, the movement of the planets, natural cycles, and
natural shapes - there are circles absolutely everywhere. The circle is one
of the most complex shapes, and indeed the most difficult for man to
create, yet nature manages to do it perfectly. The centers of flowers,
eyes, and many more things are circular and we see them in our every day
life
• An ellipse is a shape that is formed when a cylinder is cut at an angle. If you
tilt a glass of water so that the surface is no longer horizontal, the resulting
shape of the water is an ellipse. Ellipses can also be seen when a hula hoop or
tire of a car is turned askew. Though these are real life examples of optical
ellipses, the ellipse also has uses in real life.
Task 1 / Introduction2
• Parabolas are used in flashlight reflectors, satellite dish antennas,
radio telescopes, and in other devices. They are also seen in
phenomenon such as trajectory of a object having some mass and
under constant gravitational pull. basketball (or any other ball)
flying through the air,
• A hyperbola is the set of all points such that the difference of the
distances between any point on the hyperbola and two fixed points
is constant. The two fixed points are called the foci of the
hyperbola.
1
Task
/ Picture Album
1- Circle
2- Parabola
3- Ellipse
4- Hyperbola
2
Task
/ Conics Definitions
Conics
Definition
Parabola
The set of all points in a plane that are the same distance from a given point
called focus and given line called the directrix
Circle
Is the set of all points in a plane that are equidistant from a given point in
the plane, called center
Ellipse
Is the set of all points in the plane such that the sum of the distance from
two fixed points is constant
Hyperbola
Is the set of all points in a plane such that the absolute value of the
differences of the distances from the foci is constant
Task2 /Standard Forms of Equations of Conic
Sections :
Type of conics
Equations
(x – h)2 + (y – k)2 = r2
Center is (h, k).
Circle
Radius is r.
Parabola with
horizontal axis
Vertex is (h, k). Focus is (h + p,
k). Directrix is the line x = h –
p. Axis is the line y = k.
Parabola with
Vertical axis
X= a (y-k) 2 + h
Y= a (x-h) 2 + k
Vertex is (h, k). Focus is (h, k +
p). Directrix is the line y = k –
p. Axis is the line x = h.
Task2
Type of conics
Equations
Ellipse with
horizontal
major axis
Center is (h, k). Length of major axis is
2a. Length of minor axis is 2b. Distance
between center and either focus is c with c2 =
a2– b2, a > b > 0.
Ellipse with
Vertical major
axis
Center is (h, k). Length of major axis is 2a.
Length of minor axis is 2b. Distance
between center and either focus is c with c2 =
a2– b2, a > b > 0.
Hyperbola with
horizontal
transverse axis
Center is (h, k). Distance between the
vertices is 2a. Distance between the foci is
2c. c2 = a2 + b2
Hyperbola with
vertical
transverse axis
Center is (h, k). Distance between the vertices
is 2a. Distance between the foci is 2c. c2 =
a 2 + b2
Task3 /Parabola
• You can find the equation of a line by knowing two points from that
line, know to find and equation of parabola you need to know three
points.
• Find the equation of a parabola that pass through (0,3), (-2, 7) and
(1, 4). [hint: use the standard quadratic equation: Y= ax2+bx+c
Task3 /Parabola
• (0,3)(-2,7) and (1,4)
3 = a(0) + b(0) +c ~> c =
3
• 7=a(-2)2+b(-2)+3
4/2= 4a/2 -2b/2 ~> 2 = 2a – b
• 4 = a(1)2+b(1)+3
1 = a+b
| a+b=1
2 =2a - b
| 1+b=1
–––––––––
| b=0
3/3=3a/3 ~> a = 1
Check :Y=(0)2+3
Y= 3
• Y = ax2+bx+c
Y=(1)x2 +(0)x+(3)
Y= x2 +3
| (0,3)
|
Task3 /Parabola The Graph of the function: y=x2+3
Task3 /Circle
• If you have a line equation 𝑥+2𝑦=2 and circle equation 𝑥2+𝑦2 =25 .
How many points the graphs of these two equations have in
common.
• Now Graphically explore the all cases of line and circle intersections
in the plane.
Task3 /Circle
• Line equation : x+2y=2
• Circle equation : x2+y2=25
X+2y=2 ~ 2y/2=2-x/2
• Y= 1 – x/2
Point of
inersection(
x2=-4)
There will be two points of intersection
at x= -4 and x=4.8
X2 + ( 1-x/2)2 = 25
X2 + 1 – x+ ¼x2 =25
5/4x2 – x + 1 =25
5/4x2-x-24=0
• X1 = 4.8
• X2 = –4
Point of
intersection(
x1=4.8)
Task4 / Physics
• The path of any thrown ball is parabola. Suppose a ball is thrown
from ground level, reach a maximum height of 20 meters of and hits
the ground 80 meters from where it was thrown. Find the equation
of the parabolic path of the ball, assume the focus is on the ground
level.
Task4 / Physics
• Height = 20 , a = 80 => 1/-80
Focus = (0,0) , Center = (0,20)
• Y = 1/-80 (x-0) 2 + 20
Task4 / Halley's Comet
• It takes about 76 years to orbit the Sun, and since it’s path is an
ellipse so we can say that its movements is periodic. But many other
comets travel in paths that resemble hyperbolas and we see it only
once. Now if a comet follows a path that is one branch of a
hyperbola. Suppose the comet is 30 million miles farther from the
Sun than from the Earth. Determine the equation of the hyperbola
centered at the origin for the path of the comet. Hint: the foci are
Earth and the Sun with origin in the middle.
Task4 / Halley's Comet
• Hint: the foci are Earth and the Sun with
origin in the middle.
• Values of 2c =146/2 c = 73
• a = 30/2 = 15
c 2 = a 2 + b2
5329 = 225 + b2
b2 = 5104
• X2/225 – Y2/5104= 1