Each of the geometric figures are obtained
by intersecting a double-napped right
circular cone with a plane. Thus, the
figures are called conic sections or
conics.
Definition:
A conic section is the intersection of a
plane and a cone.
By changing the angle and location of
intersection, we can produce a circle,
ellipse, parabola or hyperbola
Which are generally the different ways you
can divide plane and a cone in a
symmetrical way.

If the plane cuts
completely across one
nappe of the cone and
is perpendicular to the
axis of the cone, the
curve of the section is
called a circle. AND It’s
a special case of
Ellipse which you will
see the next slide.
If the plane isn't
perpendicular to the axis of
the cone, it is called an
ellipse.
 An ellipse is the set of all
points in a plane, the sum of
the distances from two fixed
points in the plane is
constant.
 Many comets have elliptical
orbits.


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If the plane doesn't cut
across one entire nappe
or intersect both
nappes, the curve of the
intersection is called a
parabola.
A parabola is the set of
all points in a plane
equidistant from a fixed
point and a fixed line in
the plane.

If the plane cuts
through both nappes of
the cone, the curve is
called a hyperbola.
 The hyperbola is the set
of all points in a plane.
The difference of whose
distance from two fixed
points in the plane is the
positive constant.

The Greeks discovered the
properties that define conics
in terms of points and lines.
These properties are
important tool for current day
problems with the behaviors
of atoms, molecules and
outer space. Particles that
move under the influence of
an inverse square force field
has a path that is described by
conic sections.
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).
1.
(0,3)
y = ax2 + bx + c
3 = a(0)2 + b(0) + c
c=3
2.(-2, 7)
y = ax2 + bx + c
7 = a(-2)2 + b(-2) + 3
4a-2b=4
5.
3. (1, 4)
y = ax2 + bx + c
4 = a(1)2 + b(1) + 3
a+b=1
4.
4a-2b=4
a+b=1
b=1-a
4a-2(1-a)=4
4a-2+2a=4
6a=6
a=1
b=0
Equation :
y =(1)x +(0)x + c
y = x2+ 3
Circle

If you have a line equation 𝑥𝑥𝑦𝑦and circle equation 𝑥𝑥𝑦𝑦. How many
points the graphs of these two equations have in common.
x=-2y+2
(-2y+2)2 + y2 = 25
5y2-8y-21 = 0
y1=3
x1=-4
y2=-1.4
x2=4.8
(-4, 3)
(4.8, -1.4)

Now Graphically explore the all cases of line and circle
intersections in the plane.
A=(-4,3)
B=(4.8, -1.4)
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.
Vertex (0,20)
c= |1/a|=|1/80|=80
Y=a(x-h)2+k
Y=-1/80(x-0)2+20(because
it’s vertical)
Y=-1/80x2+20

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.
C=38 (76/2)
a=15 (30/2)
Final equation:
C2=a2+b2
382=152+b2
1444=225+b2
b2=1444-225
b2=1219
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Eiman Alneyadi
Alanunod yousef
Mouza Al ahbabi
Alaounod Thani
12.53