1. What is the connection?

3. Conic Sections

4. The early Greek astronomers thought that the planets moved in circular orbits about an unmoving earth, since the circle is the simplest mathematical curve.
In the 17th century, Johannes Kepler eventually discovered that each planet travels around the sun in an elliptical orbit with the sun at one of its foci.
Kepler’s Three Laws of planetary motion.

5. On a far smaller scale, the electrons of an atom move in an approximately elliptical orbit with the nucleus at one focus.

6. One of nature's best known approximations to parabolas is the path taken by a body projected upward, as in the parabolic trajectory of a golf ball.

7. The easiest way to visualize the path of a projectile is to observe a waterspout.
Each molecule of water follows the same path and, therefore, reveals a picture of the curve.

8. This discovery by Galileo in the 17th century made it possible for cannoneers to work out the kind of path a cannonball would travel if it were hurtled through the air at a specific angle.

9. Jigsaw Group Activity
You will be given a worksheet with 4 sections: circle, ellipse, hyperbola, parabola
Please form groups of 4 people
You will also be given a color card. On my prompt, you will form a new group with all of the individuals who have the same color as you
Your assignments:
pink: circle
orange: ellipse
green: Parabola
yellow: Hyperbola

12. Learning Objectives – The Parabola
Know the names of the Conics
Analyze parabolas with vertex at the origin
Analyze parabolas with vertex at (h,k)
Solve applied problems involving parabolas

13. The Parabola

14. Focus (0,p)
Vertex (h,k)
Geometric Definition of a Parabola: The collection of all the points P(x,y) in a a plane that are the same distance from a fixed point, the focus, as they are from a fixed line called the directrix. P
Directrix

15. Focus (0,p) p Vertex (h,k) p Directrix y = -p And the equation is…
As you can plainly see the distance from the focus to the vertex is “p” and is the same distance from the vertex to the directrix!
Focus (0,p) p
Vertex (h,k) p
Directrix
y = -p
And the equation is…

16. Directrix: y = p p
Vertex (h,k) p
Focus (0,-p)
And the equation is…

17. Directrix: y = -p p p
Focus (p,0)
Vertex (h,k)
And the equation is…

18. p p
Directrix: y = p
Vertex (h,k)
Focus (-p,0)
And the equation is…

19. Parabolas exhibit unusual and useful reflective properties.
If a light is placed at the focus of a parabolic mirror, the light will be reflected in rays parallel to its axis.
In this way a straight beam of light is formed.
It is for this reason that parabolic surfaces are used for headlamp reflectors.
The bulb is placed at the focus for the high beam and in front of the focus for the low beam.

20. The opposite principle is used in the giant mirrors in reflecting telescopes and in antennas used to collect light and radio waves from outer space:
...the beam comes toward the parabolic surface and is brought into focus at the focal point.

21. STANDARD FORMS
I like to call standard form “Good Graphing Form”

22. A parabola that opens up/down:
A Couple More Things…………
A parabola that opens up/down:
Has a vertex at (h , k)
Has an Axis of Symmetry at x = h
3) Coordinates of focus (h , k+p)
4) Equation of Directrix y = k-p
A parabola that opens right/left:
Has a vertex at (h , k)
Has an Axis of Symmetry at y = k
3) Coordinates of focus (h+p , k)
4) Equation of Directrix x = h-p

23. Finding the Equation of a Parabola with Vertex (0, 0)
A parabola has vertex (0, 0) and the focus on an axis.
Write the equation of each parabola.
a) The focus is (-6, 0).
b) The directrix is defined by x = 5.
c) The focus is (0, 3).

24. Finding the Equation of a Parabola with Vertex (0, 0)
A parabola has vertex (0, 0) and the focus on an axis.
Write the equation of each parabola.
a) The focus is (-6, 0).
Since the focus is (-6, 0), the equation of the parabola is y2 = 4px.
p is equal to the distance from the vertex to the focus, therefore p = -6.
The equation of the parabola is y2 = -24x.
b) The directrix is defined by x = 5.
Since the focus is on the x-axis, the equation of the parabola is y2 = 4px.
The equation of the directrix is x = -p, therefore -p = 5 or p = -5.
The equation of the parabola is y2 = -20x.
c) The focus is (0, 3).
Since the focus is (0, 3), the equation of the parabola is x2 = 4py.
p is equal to the distance from the vertex to the focus, therefore p = 3.
The equation of the parabola is x2 = 12y.

25. Finding the Equations of Parabolas
Write the equation of the parabola with a focus at (3, 5) and
the directrix at x = 9, in standard form.
For those who finish, try out this problem:
Find the equation of the parabola that has a minimum at
(-2, 6) and passes through the point (2, 8).

26. Finding the Equations of Parabolas
Write the equation of the parabola with a focus at (3, 5) and
the directrix at x = 9, in standard form.
The distance from the focus to the directrix is 6 units,
therefore, 2p = -6, p = -3. Thus, the vertex is (6, 5).
The axis of symmetry is parallel to the x-axis:
(y - k)2 = 4p(x - h)
h = 6 and k = 5
(y - 5)2 = 4(-3)(x - 6)
(y - 5)2 = -12(x - 6)
(6, 5)
Standard form

27. Finding the Equations of Parabolas
Find the equation of the parabola that has a minimum at
(-2, 6) and passes through the point (2, 8).

28. Finding the Equations of Parabolas
Find the equation of the parabola that has a minimum at
(-2, 6) and passes through the point (2, 8).
The axis of symmetry is parallel to the y-axis.
The vertex is (-2, 6), therefore, h = -2 and k = 6.
Substitute into the standard form of the equation
and solve for p:
(x - h)2 = 4p(y - k)
x = 2 and y = 8
(2 - (-2))2 = 4p(8 - 6)
16 = 8p
2 = p
(x - h)2 = 4p(y - k)
(x - (-2))2 = 4(2)(y - 6)
(x + 2)2 = 8(y - 6)
Standard form

29. Find the coordinates of the vertex and focus,
Analyzing a Parabola
Find the coordinates of the vertex and focus,
the equation of the directrix, the axis of symmetry,
and the direction of opening of y2 - 8x - 2y - 15 = 0.
4p = 8
p = 2
y2 - 8x - 2y - 15 = 0
y2 - 2y + _____ = 8x _____ 1 1
(y - 1)2 = 8x + 16
(y - 1)2 = 8(x + 2)
Standard
form
The vertex is (-2, 1).
The focus is (0, 1).
The equation of the directrix is x + 4 = 0.
The axis of symmetry is y - 1 = 0.
The parabola opens to the right.
3.6.11

30. y2 - 10x + 6y - 11 = 0 Graphing a Parabola
For those who finish early, try this out:
y2 - 10x + 6y - 21 = 0
How does this second graph relate to the first?

31. Learning Objectives Know the names of the Conics
Analyze parabolas with vertex at the origin
Analyze parabolas with vertex at (h,k)
Solve applied problems involving parabolas

32. Ellipse Definitions
Algebraic Definition of an Ellipse: a set of points in the plane such that the sum of the distances from two fixed points, called foci, remains constant.
If you are unable to do the activity, skip this slide.

33. Any cylinder sliced on an angle will reveal an ellipse in cross-section
(as seen in the Tycho Brahe Planetarium in Copenhagen).

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.

35. The principle is also used in the construction of "whispering galleries" such as in 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. Learning Objectives – The Ellipse
Analyze ellipses with center at the origin
Analyze ellipses with center at (h,k)
Solve applied problems involving ellipses

37. The Ellipse

38. The Ellipse An ellipse is the locus of all points in a plane such that
the sum of the distances from two given points in the plane,
the foci, is constant.
Minor Axis
Major Axis
Focus 1
Focus 2
Point

39. The Standard Forms of the Equation of the Ellipse
The standard form of an ellipse centered at the origin with the major
axis of length 2a along the x-axis and a minor axis of length 2b along
the y-axis, is:

40. The Standard Forms of the Equation of the Ellipse [cont’d]
The standard form of an ellipse centered at the origin with
the major axis of length 2a along the y-axis and a minor axis
of length 2b along the x-axis, is:

41. a2 = b2 + c2 b2 = a2 - c2 c2 = a2 - b2 The Pythagorean Property b a c
F1(-c, 0)
F2(c, 0)
Length of major axis: 2a
Length of minor axis: 2b
Vertices: (a, 0) and (-a, 0)
Foci: (-c, 0) and (c, 0)

42. The Standard Forms of the Equation of the Ellipse [cont’d]
The standard form of an ellipse centered at any point (h, k)
with the major axis of length 2a parallel to the x-axis and
a minor axis of length 2b parallel to the y-axis, is:
(h, k)

43. The Standard Forms of the Equation of the Ellipse [cont’d]
The standard form of an ellipse centered at any point (h, k)
with the major axis of length 2a parallel to the y-axis and
a minor axis of length 2b parallel to the x-axis, is:
(h, k)

44. Finding the Centre, Axes, and Foci
State the coordinates of the vertices, the coordinates of the foci,
and the lengths of the major and minor axes of the ellipse,
defined by each equation.

45. Finding the Centre, Axes, and Foci
State the coordinates of the vertices, the coordinates of the foci,
and the lengths of the major and minor axes of the ellipse,
defined by each equation. a)
The center of the ellipse is (0, 0).
Since the larger number occurs under the x2,
the major axis lies on the x-axis. a
The length of the major axis is 8. b
The length of the minor axis is 6. c
The coordinates of the vertices are (4, 0) and (-4, 0).
To find the coordinates of the foci, use the Pythagorean property:
c2 = a2 - b2 = =
= 7
The coordinates of the foci are:
and

46. Finding the Center, Axes, and Foci
b) 4x2 + 9y2 = 36

47. Finding the Center, Axes, and Foci
b) 4x2 + 9y2 = 36
The center of the ellipse is (0, 0).
Since the larger number occurs under the x2,
the major axis lies on the x-axis.
The length of the major axis is 6. b a
The length of the minor axis is 4. c
The coordinates of the vertices are (3, 0) and (-3, 0).
To find the coordinates of the foci, use the Pythagorean property.
c2 = a2 - b2 =
= 9 - 4
= 5
The coordinates of the foci are:
and

48. Finding the Equation of the Ellipse With Center at (0, 0)
a) Find the equation of the ellipse with center at (0, 0),
foci at (5, 0) and (-5, 0), a major axis of length 16 units,
and a minor axis of length 8 units.

49. Finding the Equation of the Ellipse With Center at (0, 0)
a) Find the equation of the ellipse with center at (0, 0),
foci at (5, 0) and (-5, 0), a major axis of length 16 units,
and a minor axis of length 8 units.
Since the foci are on the x-axis, the major axis is the x-axis.
The length of the major axis is 16 so a = 8.
The length of the minor axis is 8 so b = 4.
Standard form

50. Finding the Equation of the Ellipseb)

51. Finding the Equation of the Ellipse With Center at (0, 0) b)
The length of the major axis is 12 so a = 6.
The length of the minor axis is 6 so b = 3.
Standard form

52. Finding the Equation of the Ellipse
Find the equation for the ellipse with the center at (3, 2),
passing through the points (8, 2), (-2, 2), (3, -5), and (3, 9).
(3, 2)

53. Finding the Equation of the Ellipse With Center at (h, k)
Find the equation for the ellipse with the center at (3, 2),
passing through the points (8, 2), (-2, 2), (3, -5), and (3, 9).
The major axis is parallel to the y-axis and has a length of 14 units, so a = 7.
The minor axis is parallel to the x-axis and has a length of 10 units, so b = 5.
The center is at (3, 2), so h = 3 and k = 2.
(3, 2)
Standard form

54. Finding the Equation of the Ellipse b)
(-3, 2)

55. Finding the Equation of the Ellipse With Center at (h, k) b)
The major axis is parallel to the x-axis and
has a length of 12 units, so a = 6.
The minor axis is parallel to the y-axis and
has a length of 6 units, so b = 3.
The centre is at (-3, 2), so h = -3 and k = 2.
(-3, 2)
Standard form
3.4.14

56. Please identify the center, foci, major and minor axis:
a) x2 + 4y2 - 2x + 8y - 11 = 0

57. Analysis of the Ellipse [cont’d]
a) x2 + 4y2 - 2x + 8y - 11 = 0
x2 + 4y2 - 2x + 8y - 11 = 0
(x2 - 2x ) + (4y2 + 8y) - 11 = 0
(x2 - 2x + _____) + 4(y2 + 2y + _____) = _____ + _____ 1 1 1 4
(x - 1)2 + 4(y + 1)2 = 16
Since the larger number
occurs under the x2, the
major axis is parallel to
the x-axis.
h =
k =
a =
b = 1 -1 4 2
c2 = a2 - b2 = =
= 12
The center is at (1, -1).
The major axis, parallel to the x-axis,
has a length of 8 units.
The minor axis, parallel to the y-axis,
has a length of 4 units.
The foci are at
and
3.4.16

58. Learning Objectives – The Ellipse
Analyze ellipses with center at the origin
Analyze ellipses with center at (h,k)
Solve applied problems involving ellipses