4. Objectives:
At the end of the lesson, the student is able to:
1. Illustrate the different types of conic sections: parabola,
ellipse, circle, hyper- bola, and degenerate cases;
2. Define a circle;
3. Determine the standard form of equation of a circle;
4. Graph a circle in a rectangular coordinate system; and
5. Solve situational problems involving conic sections
(circles).
Introduction
We present the conic sections, a particular class of
curves which sometimes appear in nature and which
have applications in other fields. In this lesson, we
first illustrate how each of these curves is obtained
from the intersection of a plane and a cone, and then
discuss the first of their kind, circles. The other conic
sections will be covered in the next lessons.
5. Conic Sections
*The four basic conic
sections are all created
by cutting a double
cone at different angles.
There are 4 conic sections
• Circle
• Ellipse
• Parabola
• Hyperbola
parabol a
C1FC1P
BX1S
hgperbola
e11pse
6.
7. In "primitive" terms, a circle is the shape formed
in the surface by driving a pen (the "center") into
the surface, putting a loop of string around the
center, pulling that loop taut with another pen,
and dragging that second pen through the
surface at the further extent of the loop of string.
The resulting figure drawn in the surface is a
circle.
In algebraic terms, a circle is the set (or "locus")
of points (x, y) at some fixed distance r from some
fixed point (h, k). The value of r is called the
"radius" of the circle, and the point (h, k) is called
the "center" of the circle.
8. The Standard Form of a circle with a center at (0,0)
and a
radius, r, is x2.. + y2 =r2
Center at the origin:
X2 + Y2 = r2
C (0,0)
Radius r = 5
9. The "general" equation of a circle is:
x2 + y2 + Dx + Ey + F = 0
The "center-radius" form of the equation
is: (x – h)2 + (y – k)2 = r2
...where the h and the k come from the
center point (h, k) and the r2 comes from
the radius value r. If the center is at the
origin, so (h, k) = (0, 0), then the equation
simplifies to x2 + y2 = r2.
10. State the center and radius of the circle with the equation
(x – 2)2 + y2
= 52, and sketch the circle.
The y2 term means the same thing as (y – 0) 2, so the
equation is really (x – 2) 2 + (y – 0) 2 = 52, and the center must
be at (h, k) = (2, 0). Clearly, the radius is r = 5.
To sketch, I'll first draw the dot for
the center:
point drawn at (h, k) = (2, 0)
11.
12. A parabola is the set of all points in a plane
such that each point in the set is equidistant
from a line called the directrix and a fixed point
called the focus.
A parabola is a curve that looks like the one shown
above. Its open end can point up, down, left or right. A
curve of this shape is called 'parabolic', meaning 'like a
parabola'.
There are three common ways to define a parabola:
What's in a parabola?
13. 1. Focus and Directrix
In this definition we start with a line
(directrix) and a point (focus) and plot
the locus of all points equidistant from
each.
PARABOLA
14. 2. The graph of a function
When we plot the graph of a function of the form
the x2 term causes it to be in the shape of a parabola.
PARABOLA
15. 3. As a conic section
A parabola is formed at the intersection of a plane
and a cone when the plane is parallel to one side
of the cone.
PARABOLA
16. The parabola has many important applications,
from a parabolic antenna or parabolic microphone
to automobile headlight reflectors and the design of
ballistic missiles. They are frequently used in
physics, engineering, and many other areas.
What is the importance of parabola?
PARABOLA
17. • The Standard Form of a Parabola that opens
to the right and has a vertex at (0,0) is……
y2
= 4px
PARABOLA
18. The Parabola that opens to the right and has a vertex at
(0,0) has the following characteristics……
p is the distance from the vertex of the parabola to the
focus or directrix
This makes the coordinates of the focus (p,0)
This makes the equation of the directrix x = -p
The makes the axis of symmetry the x-axis (y = 0)
PARABOLA
19. The Standard Form of a Parabola that opens
to the left and has a vertex at (0,0) is......
y2 = -4ax
PARABOLA
20. The Parabola that opens to the left and has a vertex at
(0,0) has the following characteristics……
p is the distance from the vertex of the parabola to the
focus or directrix
This makes the coordinates of the focus(-p,0)
This makes the equation of the directrix x = p
The makes the axis of symmetry the x-axis (y = 0)
PARABOLA
21. The Parabola that opens up and has a vertex at
(0,0) has the following characteristics……
p or a is the distance from the vertex of the parabola
to the focus or directrix
This makes the coordinates of the focus (0,p)
This makes the equation of the directrix y = -p
This makes the axis of symmetry the y-axis (x = 0)
PARABOLA
22. The Standard Form of a Parabola that opens
down and has a vertex at (0,0) is……
x2
= −4py
PARABOLA
23. The Standard Form of a Parabola that opens to
the right and has a vertex at (h,k) is……
(y −k)2
= 4p(x −h)
PARABOLA
24. The Parabola that opens to the right and has a
vertex at (h,k) has the following
characteristics……..
2a
p is the distance from the vertex of the parabola
to the focus or directrix
This makes the coordinates of the focus (h+p, k)
This makes the equation of the directrix x = h – p
This makes the axis of symmetry…….
−b
y =
PARABOLA
25. The Standard Form of a Parabola that opens to the left
and has a vertex at (h,k) is……
(y −k)2
= −4p(x −h)
PARABOLA
26. The Parabola that opens to the left and has a
vertex at (h,k) has the following
characteristics……
p is the distance from the vertex of the parabola to the
focus or directrix
This makes the coordinates of the focus (h – p, k)
This makes the equation of the directrix x = h + p
The makes the axis of symmetry
2a
−by =
PARABOLA
27. The Standard Form of a Parabola that
opens up and has a vertex at (h,k) is……
(x −h)2
= 4p(y −k)
PARABOLA
28. The Parabola that opens up and has a vertex at
(h,k) has the following characteristics……
p is the distance from the vertex of the parabola to
the focus or directrix
This makes the coordinates of the focus (h , k + p)
This makes the equation of the directrix y = k – p
The makes the axis of
symmetry
PARABOLA
29. The Standard Form of a Parabola that opens down and
has a vertex at (h,k) is……
(x −h)2
= −4p(y −k)
PARABOLA
30. The Parabola that opens down and has a
vertex at (h,k) has the following
characteristics……
➢ p is the distance from the vertex of the
parabola to the focus or directrix
➢ This makes the coordinates of the
focus (h , k - p)
➢ This makes the equation of the
directrix y = k + p
➢ This makes the axis of symmetry
31.
32. Ellipse
The Quezon Memorial Circle is a national park and a
national shrine located in Quezon City. Road
surrounding the QC Circle is actually an elliptical road.
33. The set of all points in the plane, the sum of
whose distances from two fixed points,
called the foci, is a constant. (“Foci” is the
plural of “focus”, and is pronounced FOH-
sigh.)
Ellipse
What is an Ellipse?
34. The ellipse has an important property that is
used in the reflection of light and sound
waves. Any light or signal that starts at one
focus will be reflected to the other focus.
This principle is used in lithotripsy, a
medical procedure for treating kidney
stones. The patient is placed in a elliptical
tank of water, with the kidney stone at one
focus. High-energy shock waves generated
at the other focus are concentrated on the
stone, pulverizing it.
Why are the foci of Ellipse important?
35. St. Paul's Cathedral in
London. 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.
36. General Rules
➢ x and y are both squared
Equation always equals(=) 1
Equation is always plus(+)
➢ a2 is always the biggest denominator
c2 = a2 – b2
➢ c is the distance from the center to
each foci on the major axis
➢ The center is in the middle of the 2
vertices, the 2 covertices, and the 2
foci.
Ellipse
37. Ellipse
General Rules
➢ a is the distance from the center to
each vertex on the major axis
➢ b is the distance from the center to
each vertex on the minor axis
(co—vertices)
➢ Major axis has a length of 2a
➢ Minor axis has a length of 2 b
➢ Eccentricity(e): e = c/a (The closer it
gets to 1, the closer it is to being
circular)
38. General Rules
➢ a is the distance from the center to each
vertex on the major axis
➢ b is the distance from the center to each
vertex on the minor axis (co-vertices)
➢ Major axis has a length of 2a
➢ Minor axis has a length of 2b
➢ Eccentricity(e): e = c/a (The closer e gets
to 1, the closer it is to being circular)
Ellipse
39.
40.
41. The standard form of the ellipse with a
center at (0,0) and a vertical axis is……
= 1
x 2
+
y 2
b 2
a 2
Ellipse
42. The ellipse with a center at (0,0) and a vertical axis
has the following characteristic
➢ Vertices (± a,0)
➢ Co-Vertices (0, ± b)
➢ Foci ( c,0)
Ellipse
43. The standard form of the ellipse with a
center at (h,k) and a horizontal axis is……
=1
(x −h)2
+
(y −k)2
a2
b2
Ellipse
44. The ellipse with a center at (h,k) and a
horizontal axis has the following
characteristics......
➢ Vertices (h ±a , k)
➢ Co-Vertices (h, k ± b)
➢ Foci (h ± c , k)
Ellipse
45. The standard form of the ellipse with a
center at (h,k) and a vertical axis is……
1
(x−h)2
+
( y−k)2
=
b2
a2
Ellipse
46. The ellipse with a center at (h,k) and a vertical
axis has the following characteristics……
Vertices (h, k ± a)
Co-Vertices (h±b , k)
Foci (h, k ± c)
Ellipse
47.
48. The set of all points in the plane, the
difference of whose distances from
two fixed points, called the foci,
remains constant.
What is Hyperbola?
49. Where are the
Hyperbolas?
* A sonic boom shock wave has the shape of a
cone, and it intersects the ground in part of a
hyperbola. It hits every point on this curve at
the same time, so that people in different
places along the curve on the ground hear it at
the same time. Because the airplane is
moving forward, the hyperbolic curve moves
forward and eventually the boom can be
heard by everyone in its path.
50. Hyperbola
General Rules
◦ The center is in the middle of the 2 vertices
and the 2 foci.
◦ The vertices and the covertices are used to
draw the rectangles that form the
asymptotes.
◦ The vertices and the covertices are the
midpoints of the rectangle
◦ The covertices are not labeled on the
hyperbola because they are not actually part
of the graph
51.
General Rules
◦ b is the distance from the center to
each midpoint of the rectangle used to
draw the asymptotes. This distance
runs perpendicular to the distance (a).
◦ Major axis has a length of 2a
◦ Eccentricity(e):e = c/a (The closer it
gets to 1, the closer it is to being
circular
◦ If x2 is first then the hyperbola is
horizontal
◦ If y2 is first then the hyperbola is
vertical.
Hyperbola
52.
General Rules
• The center is in the middle of the
2 vertices and the 2 foci.
• The vertices and the covertices are
used to draw the rectangles that
form the asymptotes.
• The vertices and the covertices are
the midpoints of the rectangle
• The covertices are not labeled on the
hyperbola because they are not
actually part of the graph
Hyperbola
53. A basketball court where both the keys
And three point lines, are hyperbola
54.
55.
56. The standard form of the
Hyperbola with a center at (0,0)
and a vertical axis is……
= 1
y 2
−
x 2
a 2
b 2
Hyperbola
57.
58.
59.
60.
The standard form of the
Hyperbola with a center at
(h,k) and a vertical axis is……
1(y−k)2
−
(x−h)2
=
a2
b2
Hyperbola
61.
62. Conic Sections Practice Test 1.
Give the coordinates of the circle's center
and it radius. ( x − 2 ) 2 + ( y + 9 ) 2 = 1
2. Find the equation of the circle graphed
below.
A) x2 + y 2 = 4 C) x2 + y 2 = 16
B) x2 + y = 16 D) y2 = x2 + 16
E) x2 + y2 = 1
63. 3. Graph the following equation.
x 2 − 10x + y 2 = -9
Conic Sections Practice Test 1.
4. Find the vertex and focus of the parabola.
(y − 2)2 + 16(x − 3) = 0
5. Find the standard form of the equation of the
parabola with the given characteristic and vertex
at the origin. focus: (0, 7)
A) x2
= 28y C) x2
= –7y
B) y2
= 7x D) x2
= 7y
E) y2
= 28x
