1. Conic Sections

2. Table of Contents Sections of a right cone The Conics in everyday life
Terminology
Conics as plane loci - Problems
Double Hyperbola
Conics as plane loci
Four Conic Sections
Conics in a rectangle
More Problems
Tangent to the Conics 1
Focal Spheres
Tangents to the Conics 2 (HL)
Common Properties
Centre of Curvature & Evolute

3. FOUR CONIC SECTIONS

4. The circle, ellipse, parabola and hyperbola are known as conic sections
All four curves are obtained by slicing a double cone at varying angles

5. When a plane is parallel to one generator, the section is a parabola
When a plane cuts all generators on one side of the apex, the section is an ellipse
When a plane cuts both parts of the double cone, the section is a hyperbola
When a plane is parallel to one generator, the section is a parabola
When this plane is perpendicular to the axis, the section is a circle
CONICS AS SECTIONS OF A CONE
Ellipse
Circle
Parabola
Hyperbola
The parabola has only one branch and is unlimited
The hyperbola has two branches extending indefinitely in opposite directions
The ellipse and circle are closed curves 5

6. A circle is obtained when the cutting plane is perpendicular to the axis of the cone

7. An ellipse is obtained when the cutting plane cuts all elements of the upper or lower cone

8. A parabola is obtained when the cutting plane is parallel to a generating line

9. A hyperbola is obtained when the cutting plane cuts both the upper and lower cones

10. THE ELLIPSE AROUND US
The circle is the simplest of all curves
However, the ellipse is the curve most often "seen" in everyday life
This is because every circle, viewed obliquely, appears elliptical
The orbits of the planets are ellipses

11. THE PARABOLA AROUND US
When a sliothar or golf ball is hit into the air, it follows a parabolic path
The centre of gravity of a leaping salmon describes a parabola
The parabola is used by engineers in designing some suspension bridges

12. THE HYPERBOLA AROUND US
A hyperbola is formed when a right circular cone is intersected by a plane parallel to its axis
Such an intersection occurs in physical situations as simple as sharpening a pencil that has a polygonal cross section……….
…...and in the patterns formed on a floor by a flash light

13. SECTIONS OF A RIGHT CONE

14. PARABOLA
vertex
parabola
cutting plane parallel to element of cone
true shape
of cut surface
horizontal cutting plane yields a circle in plan
double ordinate

15. ELLIPSE
centre line
true shape of cut surface is an ellipse
cutting plane passing through all elements of the cone

16. ELLIPSE
File swf

17. True Shape
Of Cut Surface
Datum Line

18. cutting plane parallel to axis of cone
HYPERBOLA
cutting plane parallel to axis of cone
true shape of cut surface is a hyperbola
45°

19. FOCAL SPHERES

20. TERMINOLOGY
A focal sphere is a sphere inscribed in the cone tangential to the cutting
plane
The focus is the point of contact between the sphere and the cutting plane
The directrix is the line of intersection between the cutting plane and the
horizontal plane through the tangent circle of the sphere and cone
The axis is a line through the focus perpendicular to the directrix
The vertex is the point in which the curve intersects the axis

21. FOCAL SPHERE FOR A PARABOLA
File Parabola.EPRT

22. FOCAL SPHERE FOR A PARABOLA
directrix
vertex
Cutting plane S-S
parallel to extreme
generator of cone
focus S
vertex of
cut surface
axis
plane which contains circle of contact
between cone and sphere S
focal sphere touches cutting plane at focal point

23. FOCAL SPHERES FOR AN ELLIPSE
File - Ellipse.EPRT

24. FOCAL SPHERES FOR AN ELLIPSE
directrices
vertices
cutting plane S-S
inclined to extreme
generator of cone
focal points
axis S
planes which contain
circle of contact between
cone and sphere
focal spheres touch
cutting plane at focal points
vertices of
cut surface S

25. FOCAL SPHERE FOR A HYPERBOLA
File - Hyperbola.EPRT

26. parallel to axis of cone
FOCAL SPHERE FOR A HYPERBOLA
plane which contains circle of contact
between cone and sphere S
directrix
vertex of
cut surface
vertex
focus
cutting plane S-S
parallel to axis of cone
axis
focal sphere touches cutting plane at focal point S

27. THE CONICS IN EVERYDAY LIFE

28. THE ELLIPSE AROUND US
In an ellipse, any signal (light or sound) that starts at one focus is 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
The electrons of an atom move in an approximately elliptical orbit with the nucleus at one focus

29. THE PARABOLA AROUND US
If a light is placed at the focus of a parabolic mirror (a curved surface formed by rotating a parabola about its axis), the light will be reflected in rays parallel to the axis
This property is used in the design of flashlights and headlights
The bulb is placed at the focus for the high beam and a little above the focus for the low beam
The opposite principle is used in the giant mirrors in reflecting telescopes 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

30. THE HYPERBOLA AROUND US
The path of a comet often takes the shape of a hyperbola
Many telescopes use hyperbolic (hyperbola-shaped) lenses
Sound waves travel in hyperbolic paths, and so there are applications of the hyperbola in navigation (GPS systems)
The hyperbolic shape of water cooling towers helps the tower do its job more efficiently

31. TERMINOLOGY

32. A conic section may be defined as the locus of points in a plane whose distances from the focus are in constant ratio to their distances from the directrix. This constant is called the eccentricity.
axis
directrix F P
distance from focus F
distance from directrix
Eccentricity = PF PD =
An ellipse has eccentricity between 0 and 1
A parabola has eccentricity 1
A hyperbola has eccentricity greater than 1

33. TERMINOLOGY COMMON TO ALL CONICS
Ellipse
Eccentricity <1
Directrix F1 F2
Normal
Tangent
Chord
Minor Axis
Focal Chord
Vertex
Major Axis
Vertex
Ordinate
Minor Auxiliary Circle
Major Auxiliary Circle

34. The normal is perpendicular to the tangent
TERMINOLOGY COMMON TO ALL CONICS
Eccentricity line P
The normal is perpendicular to the tangent
tangent
normal F
axis
Latus Rectum
Double Ordinate
Parabola
Eccentricity = 1
directrix

35. TERMINOLOGY COMMON TO ALL CONICS
Double Ordinate
Latus Rectum
Transverse Axis
Axis
Vertices
Focal Points
Branches
Hyperbola
Eccentricity >1

36. ELLIPSE
CONICS AS PLANE LOCI

37. The directrix and focal point of a parabola are shown.
Locate the vertex and draw a portion of the curve
axis F
directrix

38. The eccentricity line is inclined at 45° to the axis
PARABOLA
Eccentricity = 1
The eccentricity line is inclined at 45° to the axis
The endpoints of the latus rectum give two points on the curve
All arcs are swung from F
45°
axis
vertex F
directrix

39. The directrix and focal point of a hyperbola are shown
The directrix and focal point of a hyperbola are shown. The eccentricity of the curve is Locate the vertex and draw a portion of the curve.
/4. 5
axis F
directrix

40. The endpoints of the latus rectum give two points on the curve
eccentricity line
5 units F
directrix
axis
HYPERBOLA
Eccentricity > 1
The eccentricity line is inclined at an angle greater than 45° to the axis
The endpoints of the latus rectum give two points on the curve
All arcs are swung from F
45°
vertex
4 units

41. The directrix and focal point of an ellipse are shown
The directrix and focal point of an ellipse are shown. The eccentricity of the curve is Locate the vertex and draw a portion of the curve.
/4. 3
axis F
directrix

42. The endpoints of the latus rectum give two points on the curve
3 units
eccentricity line F
directrix
The endpoints of the latus rectum give two points on the curve
45°
45°
axis
VV = major axis V V
4 units
ELLIPSE
Eccentricity < 1
The eccentricity line is inclined at an angle less than 45° to the axis
All arcs are swung from F

43. The eccentricity line passes through the top of the major circle
directrix
ALTERNATE SOLUTION
3 units
eccentricity line
The eccentricity line passes through the top of the major circle
major circle
minor circle
The endpoints of the latus rectum give two points on the curve
45°
45°
axis
VV = major axis V V
4 units
ELLIPSE
Eccentricity < 1
The eccentricity line is inclined at an angle less than 45° to the axis

44. CONICS AS PLANE LOCI - PROBLEMS

45. When a plane goes faster than the speed of sound, a sonic boom shock wave occurs. A cone shaped wave shoots out of the back of the plane as shown in the photograph over and intersects the ground in part of a hyperbola (below). 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. D V
axis
In the drawing above the line DD represents the directrix of a hyperbola and V is the vertex of the curve. The eccentricity is 1.25.
Locate the focal point of the hyperbola and draw a portion of the curve.

46. When a pencil that has a hexagonal cross section is sharpened, a hyperbola is formed. This results from the intersection of the conical point of the pencil by one of its flat sides.
Two lines VF and FD are shown in the drawing below. V is the vertex of a hyperbola, F is a focal point of the curve and D is a point on the directrix.
Locate the directrix and eccentricity line and draw a portion of the curve. D F V

47. A photograph of a Cleveland Launcher driver is shown
A photograph of a Cleveland Launcher driver is shown. The underside of the club is in the shape of a parabola.
In the drawing below, F is the focus of the parabola and the line AFB represents a focal chord of the curve.
(a) Determine the position of the directrix and the axis of the parabola.
(b) Draw the portion of the curve as shown.

48. ELLIPSE
CONICS IN A RECTANGLE

49. PARABOLA
vertex
The same number of equal divisions must be used for each half
axis
4 equal divisions
4 equal divisions

50. ELLIPSE
5 equal divisions
minor axis
major axis
5 equal divisions
The same number of equal divisions must be used for each quadrant

51. HYPERBOLA axis V1 V2 5 equal divisions
The same number of equal divisions are used
5 equal divisions

52. MORE PROBLEMS

53. A photograph of a Parabola Hall Table designed by Nathan Hunter is shown. Two curly wood panels intersect steel parabolas to support a glass top. The curved panels are portions of an ellipse.
The incomplete elevation of the table is given over. The two parabolas are to be inscribed in the rectangles ABCD and EFGH, respectively. P is a point on the curve of the ellipse whose minor axis MN is drawn.
Complete the view of the hall table showing all construction lines.

54. The St. Louis Science Centre Planetarium's exterior curved surface is in the shape of a hyperboloid. This shape is formed by revolving a hyperbola around its axis.
The drawing below shows an incomplete elevation of the planetarium. The line VV represents the transverse axis of the double hyperbola and P and Q are points on the curve.
Complete the elevation by drawing the double hyperbola.

55. TANGENTS TO THE CONICS 1

56. The tangent is perpendicular to the normal
TANGENT TO A CIRCLE FROM A POINT ON THE CURVE
An ellipse is a stretched circle……..
The tangent is perpendicular to the normal
90° P
normal
tangent F 1 F 2
P is joined to the foci
…... and a circle is an ellipse with the two focal points located at the centre

57. TANGENT TO AN ELLIPSE FROM A POINT P ON THE CURVE
normal F 1 F 2
The normal is perpendicular to the tangent and can be located by bisecting angle F1PF2

58. The normal is perpendicular to the tangent
TANGENT TO A PARABOLA FROM A POINT P ON THE CURVE
to other focal point
at infinity P
tangent
normal
axis F
The normal is perpendicular to the tangent
directrix

59. TANGENT TO A HYPERBOLA FROM A POINT P ON THE CURVE
normal
axis F 1 V 1 V 2 F 2
tangent

60. TANGENTS TO THE CONICS 2 (HIGHER LEVEL)

61. TANGENT TO A PARABOLA FROM A POINT P OUTSIDE THE CURVE
axis
directrix F P
to other focal point
at infinity
point of contact
tangent R
Two tangents can be drawn from P
to other focal point
at infinity

62. TANGENT TO AN ELLIPSE FROM A POINT P OUTSIDE THE CURVE
directrix P F 1 2
point of contact R
tangent
point of contact
A second tangent can be drawn from P

63. TANGENT TO A HYPERBOLA FROM A POINT P OUTSIDE THE CURVE
Radius is equal to length of transverse axis
tangent
point of contact V 2 1 F
axis P

64. DOUBLE HYPERBOLA FROM FOCI AND TRANSVERSE AXIS

65. Draw a portion of the curve when given the focal points and
a point P on the curve P
V1 – V2
a b c d e f F1 V1 V2 F2
R.V2 - c
R.V1 - c

66. Draw a portion of the curve when given the focal points and
the transverse axis.
R.V1 – d
R.V2 - d
a b c d e f F1 V1 V2 F2
R.V2 - c
R.V1 - c

67. How to locate the asymptotes and directrices for a double hyperbola
Conjugate axis
Transverse axis R
axis F1 V1 V2 F2
Auxiliary circle
directrices
asymptotes

68. PROPERTIES COMMON TO ALL CONICS

69. This property holds for all conics
The angle subtended at the focus by that part of the tangent between the focus and the directrix is 90°
point of contact
directrix F 1 2
tangent
90°

70. Perpendiculars from the foci to the tangent meet the tangent on the major auxiliary circle
major circle
tangent
axis F 1 2

71. V1 V2
axis F 1 2
In a double hyperbola, perpendiculars from the foci to the tangent meet the tangent on the auxiliary circle
tangent
auxiliary circle

72. In a parabola, a perpendicular from the focus to the tangent meets the tangent on the auxiliary circle
axis
directrix F
tangent
In a parabola, the major auxiliary circle is the tangent at the vertex

73. The line joining the midpoints of parallel chords passes through the centre of the conic
two parallel chords F 1 2
centre of ellipse

74. The line joining the midpoints of parallel chords of a parabola is parallel to the axis
directrix F
to the centre of the parabola at infinity
two parallel chords

75. CENTRE OF CURVATURE & EVOLUTE

76. CENTRE OF CURVATURE FOR A POINT P ON A PARABOLA
axis
directrix F P
to second focal point
normal
90°
centre of curvature

77. CENTRE OF CURVATURE FOR A POINT P ON AN ELLIPSE1 2
normal
centre of curvature

78. CENTRE OF CURVATURE AT THE VERTEX OF A PARABOLA
axis
directrix F V C
centre of curvature R
circle of curvature

79. The distance between V and F is added
CENTRE OF CURVATURE AT THE VERTEX OF AN ELLIPSE F 1 2 VF
The distance between V and F is added R V
centre of curvature

80. The distance between V and F is added
CENTRE OF CURVATURE AT THE VERTEX OF A HYPERBOLA F 2 1 V
The distance between V and F is added VF R
centre of curvature

81. The evolute of a conic is the locus of its centre of curvature
To construct an evolute:
Determine the centres of curvatures for a number of points on the curve
The points should include the vertices of the conic
Plot a curve through them

82. EVOLUTE OF AN ELLIPSE F 1 2 P
centre of curvature

83. The evolute of an ellipse looks like this………1 2

84. The evolute of a parabola looks like this………
axis F
Find out yourself what the evolute of a hyperbola looks like!