Math Project
Done by:
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Presentation Name
Abdulrahman Ahmed Almansoori
Mohammed Essa
Suleiman Mohammed
Saeed Ahmed Alali
Task 1
• Write an introduction about each conic section, showing what they are used for.
 Parabola:
 Circle:
A parabola can be defined as the set of
all points in a plane that are the same
distance from a given point called the
focus and a given line called the
directrix.
The line segment through the focus of a
parabola and perpendicular to the axis
of symmetry is called the latus rectum.
The endpoint of the latus rectum lie on
the parabola.
A circle is the set of all points that
are equidistant from a given point
in the plane, called the center.
Any segment with endpoints at
the center and a point on the
circle is a radius of the circle.
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Task 1
 ellipse:
 Hyperbola:
An ellipse is the set of all points in a
plane such the sum of the distances
from two fixed points is constant. These
two points are called the foci of the
ellipse. Every ellipse has two axes of
symmetry, the major axis and minor
axis. The axes are perpendicular at the
center of ellipse. The foci of an ellipse
always lie on the major axis. The
endpoints of the major axis are the
vertices of the ellipse and the endpoints
of the minor axis are the co-vertices of
the ellipse.
Similar to an ellipse, a hyperbola is the
set o all point in plane such that
absolute value of the differences of the
distances from the foci is constant. Every
hyperbola has two axes of symmetry,
the transverse axis and conjugate axis.
The axes are perpendicular at the center
of hyperbola. . The foci of an hyperbola
always lie on the transverse axis. The
vertices are the endpoints of the
transverse axis. The co-vertices are the
endpoints of the conjugate axis.
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Task 1
• Gallery: Go home, library, mall or other public places to find or take pictures for at least a
picture for an item that represent each conic section then make a picture album:
Parabola:
Circle
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Task 1
• Gallery: Go home, library, mall or other public places to find or take pictures for at least a
picture for an item that represent each conic section then make a picture album:
Ellipse
Hyperbola
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Task 2
 Comprehensive comparison between conics:
Q1) Construct a table that shows the similarities and differences between all types of conics?
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Comprehensive comparison between conics
Showing the following points:
Closed or open curve.
Its definition
Its equation
Relation between its center and focus (foci)
Other properties
Graph it
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Task 2 Answers
• Q2)
 For circle:
Shape
Closed or open curve
Definition
Equation
Relation between it’s center and focus
Other properties
Closed curve
The set of all points in a plane that are a given distance from
a given point, the center.
(x-a)2 + (y-b)2 =r2
The center and the focus is in the same place
Cutting a circular cone with a plane perpendicular to the
symmetry axis of the cone forms a circle. This intersection is
a closed curve, and the intersection is parallel to the plane
generating the circle of the cone. A circle is also the set of all
points that are equally distant from the center.
Graph
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Task 2
 For Ellipse :
Shape
Closed or open curve
Definition
Closed curve
A plane curve that results from the intersection of a cone by
a plane in a way that produces a closed curve
Equation
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Relation between it’s center and focus
where (h,k) is the center of the ellipse, rx is the distance from
the center of the circle in the x direction and ry is the
distance from the center in the y direction.
Other properties
From the center of the ellipse on the major axis. The major
axis is the line of the ellipse that has the biggest distance
from the center of the circle. If the major axis is horizontal,
2rx is the length and c2=rx2-ry2. If the major axis is vertical,
2ry is the length and c2=ry2-rx2.
Graph
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Task 2
 For Parabola :
Shape
Closed or open curve
Definition
Open curve
Equation
y = ax2 + bx + c OR x = ay2 + by + c
A conic section, created from the intersection of a right
circular conical surface and a plane parallel to a generating
straight line of that surface
Relation between it’s center and focus
Other properties
The standard equation depends on the axis of symmetry. A
vertical axis has a focus at (h,k+p) and the equation (xh)2=4p(y-k). A horizontal axis has a focus at (h+p,k) and the
equation (y-k)2=4p(x-h). The vertex is always halfway in
between the focus and directrix at a distance p from both.
Graph
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Task 2
 For Hyperbola :
Shape
Closed or open curve
Definition
Open curve
A type of smooth curve, lying in a plane, defined by its
geometric properties or by equations for which it is the
solution set.
Equation
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Relation between it’s center and focus
The foci in an hyperbola are further from the hyperbola's
center than are its vertices
Other properties
Where (h,k) is the center between the curves and it's two
asymptotes go through the points (+a,-b) and (-a,+b) as well
as (a,b) and (-a,-b) starting at the center point.
Graph
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Task 3
1. 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].
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First we get C.
3=0+0+C
C=3
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Then we get the two other equations by replacing C:
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7 = a (-2)2 + b (-2) + 3
4 = 4a – 2b
2 = 2a - b
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4 = a (1)2 + b(1) + 3
1=a+b
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2 = 2a - b
1=a+b
3 = 3a
a=1,b=0
4/2 = (4a-2b)/2
2 = 2a – b
1= a + b
3 = 3a
a=1,b=0
So the equation is going to be:
y= x2+3
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Task 3
2. 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
x + 2y = 2
2y = 2 – x
y = 1 – 0.5 x
𝑥2 + 𝑦2 = 25
x2 + (1 – 0.5 x)2 = 25
x2 + (1 – x + 0.25 x2 ) = 25
1.25 x2 – x = 24
1.25 x 2 – x – 24 = 0
• To get Y1 and Y2 we
will replace X1 and X2
in the following
equation
• y = 1 – 0.5 x, so Y1=1.4 Y2=3
1.25 x 2 – x – 24 = 0
(x+4.8) (x+4) = 0
x = -4.8 , or x = -4
• So the intersection
points are (4.8,-1.4)
and (-4,3)
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Task 4
• 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.
Vertices (0,20) and the Focus at (0,0)
It is vertical , k is different in the vertex and focus.
So the equation of the parabola is y=a(x-h)2+k
y=a(x-h)2+k
Latus rictum = I 1/a I
I 1/a I = 80
a = -1/80, because it
is open to down
y = a(x-h)2 + k
y = -180(x-0)2 + 20
y = -1/80x2 + 20
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Task 4
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.
• distance between the sun and the earth = 146 millions Km
• Centre is (0,0)
• c = 146/2 = 73
• The difference of the distances from the comet to each body is 30.
• a = 302 = 15 million miles
c2 = a2 + b2
b2 = c2 - a2
b2 = (73)2 - (15)2
b2 = 5104
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The End
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