1. Conic Sections

2. Conic Sections
One way to study a solid is to slice it open.

3. Conic Sections
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.

4. Conic Sections
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
A right circular cone

5. Conic Sections
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
A right circular cone and conic sections (wikipedia “Conic Sections”)

6. Conic Sections
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
A Horizontal Section
A right circular cone and conic sections (wikipedia “Conic Sections”)

7. Conic Sections
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
A Horizontal Section
A right circular cone and conic sections (wikipedia “Conic Sections”)

8. Conic Sections
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
A Moderately
Tilted Section
A right circular cone and conic sections (wikipedia “Conic Sections”)

9. Conic Sections
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
A Moderately
Tilted Section
A right circular cone and conic sections (wikipedia “Conic Sections”)

10. Conic Sections
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
A Horizontal Section
A Moderately
Tilted Section
Circles and
ellipsis are
enclosed.
A right circular cone and conic sections (wikipedia “Conic Sections”)

11. Conic Sections
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
A Parallel–Section
A right circular cone and conic sections (wikipedia “Conic Sections”)

12. Conic Sections
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
A Parallel–Section
A right circular cone and conic sections (wikipedia “Conic Sections”)

13. Conic Sections
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
An Cut-away
Section
A right circular cone and conic sections (wikipedia “Conic Sections”)

14. Conic Sections
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
An Cut-away
Section
A right circular cone and conic sections (wikipedia “Conic Sections”)

15. Conic Sections
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
A Horizontal Section
A Moderately
A Parallel–Section
Tilted Section
An Cut-away
Circles and Section
ellipsis are
enclosed.
Parabolas and
hyperbolas are open.
A right circular cone and conic sections (wikipedia “Conic Sections”)

16. Conic Sections
We summarize the four types of conics sections here.
Circles Ellipses
Parabolas Hyperbolas

17. Conic Sections
We summarize the four types of conics sections here.
Circles Ellipses
Parabolas Hyperbolas
Besides their differences in visual appearance and the manners
they reside inside the cone, there are many reasons, that have
nothing to do with cones, that the conic sections are grouped
into four groups.

18. Conic Sections
We summarize the four types of conics sections here.
Circles Ellipses
Parabolas Hyperbolas
Besides their differences in visual appearance and the manners
they reside inside the cone, there are many reasons, that have
nothing to do with cones, that the conic sections are grouped
into four groups. One way is to use distance relations to classify
them.

19. Conic Sections
We summarize the four types of conics sections here.
Circles Ellipses
Parabolas Hyperbolas
Besides their differences in visual appearance and the manners
they reside inside the cone, there are many reasons, that have
nothing to do with cones, that the conic sections are grouped
into four groups. One way is to use distance relations to classify
them. We use the circles and the ellipsis as examples.

20. Circles
Given a fixed point C,
C

21. Circles
Given a fixed point C, a circle is the
set of points whose distances to C is r r
a fixed constant r.
C

22. Circles
Given a fixed point C, a circle is the
set of points whose distances to C is r r
a fixed constant r.
C

23. Circles
Given a fixed point C, a circle is the
set of points whose distances to C is r r
a fixed constant r.
C

24. Circles
Given a fixed point C, a circle is the
set of points whose distances to C is r r
a fixed constant r.
C
Hence a dog tied to a
post would mark off
a circular track.

25. Circles
Given a fixed point C, a circle is the
set of points whose distances to C is r r
a fixed constant r.
The equal-distance r is called the C
radius and the point C is called the
center of the circle.

26. Circles
Given a fixed point C, a circle is the
set of points whose distances to C is r r
a fixed constant r.
The equal-distance r is called the C
radius and the point C is called the
center of the circle.
Given two fixed points (called foci),
F1 F2

27. Circles
Given a fixed point C, a circle is the
set of points whose distances to C is r r
a fixed constant r.
The equal-distance r is called the C
radius and the point C is called the
center of the circle.
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.
F1 F2

28. Circles
Given a fixed point C, a circle is the
set of points whose distances to C is r r
a fixed constant r.
The equal-distance r is called the C
radius and the point C is called the
center of the circle.
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.
Q
For example, if P, Q, and R P
are points on a ellipse,
F1 F2
R

29. Circles
Given a fixed point C, a circle is the
set of points whose distances to C is r r
a fixed constant r.
The equal-distance r is called the C
radius and the point C is called the
center of the circle.
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.
Q
For example, if P, Q, and R P
are points on a ellipse, then p2
p1
p1 + p2
F1 F2
R

30. Circles
Given a fixed point C, a circle is the
set of points whose distances to C is r r
a fixed constant r.
The equal-distance r is called the C
radius and the point C is called the
center of the circle.
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.
Q
For example, if P, Q, and R P
p2 q1
are points on a ellipse, then q2
p1
p1 + p2
F1 F2
= q1 + q2
R

31. Circles
Given a fixed point C, a circle is the
set of points whose distances to C is r r
a fixed constant r.
The equal-distance r is called the C
radius and the point C is called the
center of the circle.
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.
Q
For example, if P, Q, and R P
p2 q1
are points on a ellipse, then q2
p1
p1 + p2
F1 F2
= q1 + q2
= r1 + r 2 r1 r2
= a constant
R

32. Circles
Given a fixed point C, a circle is the
set of points whose distances to C is r r
a fixed constant r.
The equal-distance r is called the C
radius and the point C is called the
center of the circle.
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.
Q
For example, if P, Q, and R P
p2 q1
are points on a ellipse, then q2
p1
p1 + p2
F1 F2
= q1 + q2
= r1 + r 2 r1 r2
= a constant
Hence a dog leashed by a ring R
to two posts would mark off
an elliptical track.

33. Circles
Given a fixed point C, a circle is the
set of points whose distances to C is r r
a fixed constant r.
The equal-distance r is called the C
radius and the point C is called the
center of the circle.
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.
Q
For example, if P, Q, and R P
p2 q1
are points on a ellipse, then q2
p1
p1 + p2
F1 F2
= q1 + q2
= r1 + r 2 r1 r2
= a constant
Likewise parabolas and hyperbolas R
may be defined using relations of distance measurements.

34. Conic Sections
The second reason that we group the conic sections into four
types is algebraic, i.e. the equations related to graphs of the
conic sections can easily be sorted into the above four types

35. Conic Sections
The second reason that we group the conic sections into four
types is algebraic, i.e. the equations related to graphs of the
conic sections can easily be sorted into the above four types
Recall that straight lines
are the graphs of
1st degree equations
Ax + By = C
where A, B, C, are numbers.

36. Conic Sections
The second reason that we group the conic sections into four
types is algebraic, i.e. the equations related to graphs of the
conic sections can easily be sorted into the above four types
Recall that straight lines
are the graphs of
1st degree equations
Ax + By = C y = –1 x=1 y+x=1
Linear graphs
where A, B, C, are numbers.

37. Conic Sections
The second reason that we group the conic sections into four
types is algebraic, i.e. the equations related to graphs of the
conic sections can easily be sorted into the above four types
Recall that straight lines
are the graphs of
1st degree equations
Ax + By = C y = –1 x=1 y+x=1
Linear graphs
where A, B, C, are numbers.
Conic sections are the graphs of 2nd degree equations in
x and y.

38. Conic Sections
The second reason that we group the conic sections into four
types is algebraic, i.e. the equations related to graphs of the
conic sections can easily be sorted into the above four types
Recall that straight lines
are the graphs of
1st degree equations
Ax + By = C y = –1 x=1 y+x=1
Linear graphs
where A, B, C, are numbers.
Conic sections are the graphs of 2nd degree equations in
x and y. In particular, the conic sections that are parallel to the
axes (not tilted) have equations of the form
Ax2 + By2 + Cx + Dy = E, where A, B, C, D, and E are numbers.

39. Conic Sections
The second reason that we group the conic sections into four
types is algebraic, i.e. the equations related to graphs of the
conic sections can easily be sorted into the above four types
Recall that straight lines
are the graphs of
1st degree equations
Ax + By = C y = –1 x=1 y+x=1
Linear graphs
where A, B, C, are numbers.
Conic sections are the graphs of 2nd degree equations in
x and y. In particular, the conic sections that are parallel to the
axes (not tilted) have equations of the form
Ax2 + By2 + Cx + Dy = E, where A, B, C, D, and E are numbers.
The algebraic technique that enables us to sort these 2nd
degree equations into four groups of conic sections is called
"completing the square".

40. Conic Sections
The second reason that we group the conic sections into four
types is algebraic, i.e. the equations related to graphs of the
conic sections can easily be sorted into the above four types
Recall that straight lines
are the graphs of
1st degree equations
Ax + By = C y = –1 x=1 y+x=1
Linear graphs
where A, B, C, are numbers.
Conic sections are the graphs of 2nd degree equations in
x and y. In particular, the conic sections that are parallel to the
axes (not tilted) have equations of the form
Ax2 + By2 + Cx + Dy = E, where A, B, C, D, and E are numbers.
The algebraic technique that enables us to sort these 2nd
degree equations into four groups of conic sections is called
"completing the square". We will apply this method to the
circles but only summarize the results about the other ones.

41. Circles
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
r
r
center

42. Circles
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
The radius and the center completely determine the circle.
r
r
center

43. Circles
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
The radius and the center completely determine the circle.
Let (h, k) be the center of a
circle and r be the radius.
(h, k) r

44. Circles
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
The radius and the center completely determine the circle.
Let (h, k) be the center of a
circle and r be the radius.
Suppose (x, y) is a point on (x, y)
the circle, then the distance
between (x, y) and the center (h, k) r
is r.

45. Circles
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
The radius and the center completely determine the circle.
Let (h, k) be the center of a
circle and r be the radius.
Suppose (x, y) is a point on (x, y)
the circle, then the distance
between (x, y) and the center (h, k) r
is r. Hence,
r = √ (x – h)2 + (y – k)2

46. Circles
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
The radius and the center completely determine the circle.
Let (h, k) be the center of a
circle and r be the radius.
Suppose (x, y) is a point on (x, y)
the circle, then the distance
between (x, y) and the center (h, k) r
is r. Hence,
r = √ (x – h)2 + (y – k)2
or
r2 = (x – h)2 + (y – k)2

47. Circles
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
The radius and the center completely determine the circle.
Let (h, k) be the center of a
circle and r be the radius.
Suppose (x, y) is a point on (x, y)
the circle, then the distance
between (x, y) and the center (h, k) r
is r. Hence,
r = √ (x – h)2 + (y – k)2
or
r2 = (x – h)2 + (y – k)2
This is called the standard form of circles.

48. Circles
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
The radius and the center completely determine the circle.
Let (h, k) be the center of a
circle and r be the radius.
Suppose (x, y) is a point on (x, y)
the circle, then the distance
between (x, y) and the center (h, k) r
is r. Hence,
r = √ (x – h)2 + (y – k)2
or
r2 = (x – h)2 + (y – k)2
This is called the standard form of circles. Given an equation
of this form, we can easily identify the center and the radius.

49. Circles
r2 = (x – h)2 + (y – k)2

50. Circles
must be “ – ”
r2 = (x – h)2 + (y – k)2

51. Circles
r is the radius must be “ – ”
r2 = (x – h)2 + (y – k)2

52. Circles
r is the radius must be “ – ”
r2 = (x – h)2 + (y – k)2
(h, k) is the center

53. Circles
r is the radius must be “ – ”
r2 = (x – h)2 + (y – k)2
(h, k) is the center
Example A. Write the equation (–1, 8)
of the circle as shown.
(–1, 3)

54. Circles
r is the radius must be “ – ”
r2 = (x – h)2 + (y – k)2
(h, k) is the center
Example A. Write the equation (–1, 8)
of the circle as shown.
The center is (–1, 3) and the
radius is 5.
(–1, 3)

55. Circles
r is the radius must be “ – ”
r2 = (x – h)2 + (y – k)2
(h, k) is the center
Example A. Write the equation (–1, 8)
of the circle as shown.
The center is (–1, 3) and the
radius is 5.
(–1, 3)
Hence the equation is:
52 = (x – (–1))2 + (y – 3)2

56. Circles
r is the radius must be “ – ”
r2 = (x – h)2 + (y – k)2
(h, k) is the center
Example A. Write the equation (–1, 8)
of the circle as shown.
The center is (–1, 3) and the
radius is 5.
(–1, 3)
Hence the equation is:
52 = (x – (–1))2 + (y – 3)2
or
25 = (x + 1)2 + (y – 3 )2

57. Circles
r is the radius must be “ – ”
r2 = (x – h)2 + (y – k)2
(h, k) is the center
Example A. Write the equation (–1, 8)
of the circle as shown.
The center is (–1, 3) and the
radius is 5.
(–1, 3)
Hence the equation is:
52 = (x – (–1))2 + (y – 3)2
or
25 = (x + 1)2 + (y – 3 )2
In particular a circle centered at
the origin has an equation of
the form x2 + y2 = r2

58. Circles
Example B. Identify the center and
the radius of 16 = (x – 3)2 + (y + 2)2.
Label the top, bottom, left and right
most points. Graph it.

59. Circles
Example B. Identify the center and
the radius of 16 = (x – 3)2 + (y + 2)2.
Label the top, bottom, left and right
most points. Graph it.
Put 16 = (x – 3)2 + (y + 2)2 into the
standard form:
42 = (x – 3)2 + (y – (–2))2

60. Circles
Example B. Identify the center and
the radius of 16 = (x – 3)2 + (y + 2)2.
Label the top, bottom, left and right
most points. Graph it.
Put 16 = (x – 3)2 + (y + 2)2 into the
standard form:
42 = (x – 3)2 + (y – (–2))2
Hence r = 4, center = (3, –2)

61. Circles
Example B. Identify the center and
the radius of 16 = (x – 3)2 + (y + 2)2. (3, 2)
Label the top, bottom, left and right
most points. Graph it.
Put 16 = (x – 3)2 + (y + 2)2 into the (--1, --2) (7, --2)
standard form: (3, --2)
42 = (x – 3)2 + (y – (–2))2
(3, --6)
Hence r = 4, center = (3, –2)

62. Circles
Example B. Identify the center and
the radius of 16 = (x – 3)2 + (y + 2)2. (3, 2)
Label the top, bottom, left and right
most points. Graph it.
Put 16 = (x – 3)2 + (y + 2)2 into the (--1, --2) (7, --2)
standard form: (3, --2)
42 = (x – 3)2 + (y – (–2))2
(3, --6)
Hence r = 4, center = (3, –2)
When equations are not in the standard form, we have to
rearrange them into the standard form. We do this by
"completing the square".

63. Circles
Example B. Identify the center and
the radius of 16 = (x – 3)2 + (y + 2)2. (3, 2)
Label the top, bottom, left and right
most points. Graph it.
Put 16 = (x – 3)2 + (y + 2)2 into the (--1, --2) (7, --2)
standard form: (3, --2)
42 = (x – 3)2 + (y – (–2))2
(3, --6)
Hence r = 4, center = (3, –2)
When equations are not in the standard form, we have to
rearrange them into the standard form. We do this by
"completing the square". To complete the square means to
add a number to an expression so the sum is a perfect
square.

64. Circles
Example B. Identify the center and
the radius of 16 = (x – 3)2 + (y + 2)2. (3, 2)
Label the top, bottom, left and right
most points. Graph it.
Put 16 = (x – 3)2 + (y + 2)2 into the (--1, --2) (7, --2)
standard form: (3, --2)
42 = (x – 3)2 + (y – (–2))2
(3, --6)
Hence r = 4, center = (3, –2)
When equations are not in the standard form, we have to
rearrange them into the standard form. We do this by
"completing the square". To complete the square means to
add a number to an expression so the sum is a perfect
square. This procedure is the main technique in dealing with
2nd degree equations.

65. Circles
The Completing the Square Method

66. Circles
The Completing the Square Method
If we are given x2 + bx, then adding (b/2)2 to the expression
makes the expression a perfect square,

67. Circles
The Completing the Square Method
If we are given x2 + bx, then adding (b/2)2 to the expression
makes the expression a perfect square, i.e. x2 + bx + (b/2)2
is the perfect square (x + b/2)2.

68. Circles
The Completing the Square Method
If we are given x2 + bx, then adding (b/2)2 to the expression
makes the expression a perfect square, i.e. x2 + bx + (b/2)2
is the perfect square (x + b/2)2.
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2

69. Circles
The Completing the Square Method
If we are given x2 + bx, then adding (b/2)2 to the expression
makes the expression a perfect square, i.e. x2 + bx + (b/2)2
is the perfect square (x + b/2)2.
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2

70. Circles
The Completing the Square Method
If we are given x2 + bx, then adding (b/2)2 to the expression
makes the expression a perfect square, i.e. x2 + bx + (b/2)2
is the perfect square (x + b/2)2.
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2

71. Circles
The Completing the Square Method
If we are given x2 + bx, then adding (b/2)2 to the expression
makes the expression a perfect square, i.e. x2 + bx + (b/2)2
is the perfect square (x + b/2)2.
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
b. y2 + 12y + (12/2)2

72. Circles
The Completing the Square Method
If we are given x2 + bx, then adding (b/2)2 to the expression
makes the expression a perfect square, i.e. x2 + bx + (b/2)2
is the perfect square (x + b/2)2.
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
b. y2 + 12y + (12/2)2

73. Circles
The Completing the Square Method
If we are given x2 + bx, then adding (b/2)2 to the expression
makes the expression a perfect square, i.e. x2 + bx + (b/2)2
is the perfect square (x + b/2)2.
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2

74. Circles
The Completing the Square Method
If we are given x2 + bx, then adding (b/2)2 to the expression
makes the expression a perfect square, i.e. x2 + bx + (b/2)2
is the perfect square (x + b/2)2.
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2
The following are the steps in putting a 2nd degree equation
into the standard form.

75. Circles
The Completing the Square Method
If we are given x2 + bx, then adding (b/2)2 to the expression
makes the expression a perfect square, i.e. x2 + bx + (b/2)2
is the perfect square (x + b/2)2.
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2
The following are the steps in putting a 2nd degree equation
into the standard form.
1. Group the x2 and the x-terms together, group the y2 and y
terms together, and move the number term to the other
side of the equation.

76. Circles
The Completing the Square Method
If we are given x2 + bx, then adding (b/2)2 to the expression
makes the expression a perfect square, i.e. x2 + bx + (b/2)2
is the perfect square (x + b/2)2.
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2
The following are the steps in putting a 2nd degree equation
into the standard form.
1. Group the x2 and the x-terms together, group the y2 and y
terms together, and move the number term to the other
side of the equation.
2. Complete the square for the x-terms and for the y-terms.
Make sure to add the necessary numbers to both sides.

77. Circles
Example E. Use completing the square to find the center
and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,
left and right most points. Graph it.

78. Circles
Example E. Use completing the square to find the center
and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:

79. Circles
Example E. Use completing the square to find the center
and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2 – 6x + + y2 + 12y + = –36

80. Circles
Example E. Use completing the square to find the center
and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2 – 6x + + y2 + 12y + = –36 complete the squares
x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36

81. Circles
Example E. Use completing the square to find the center
and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2 – 6x + + y2 + 12y + = –36 complete the squares
x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36

82. Circles
Example E. Use completing the square to find the center
and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2 – 6x + + y2 + 12y + = –36 complete the squares
x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36
( x – 3 )2 + (y + 6)2 = 9

83. Circles
Example E. Use completing the square to find the center
and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2 – 6x + + y2 + 12y + = –36 complete the squares
x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36
( x – 3 )2 + (y + 6)2 = 9
( x – 3 )2 + (y + 6)2 = 32

84. Circles
Example E. Use completing the square to find the center
and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2 – 6x + + y2 + 12y + = –36 complete the squares
x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36
( x – 3 )2 + (y + 6)2 = 9
( x – 3 )2 + (y + 6)2 = 32
Hence the center is (3, –6),
and radius is 3.

85. Circles
Example E. Use completing the square to find the center
and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2 – 6x + + y2 + 12y + = –36 complete the squares
x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36
( x – 3 )2 + (y + 6)2 = 9
( x – 3 )2 + (y + 6)2 = 32
Hence the center is (3, –6),
and radius is 3.
(3, –6),

86. Circles
Example E. Use completing the square to find the center
and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2 – 6x + + y2 + 12y + = –36 complete the squares
x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36
( x – 3 )2 + (y + 6)2 = 9
( x – 3 )2 + (y + 6)2 = 32
Hence the center is (3, –6),
(3, –3),
and radius is 3.
(0, –6), (6, –6),
(3, –6),
(–9, –6)

87. Circles
Example E. Use completing the square to find the center
and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2 – 6x + + y2 + 12y + = –36 complete the squares
x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36
( x – 3 )2 + (y + 6)2 = 9
( x – 3 )2 + (y + 6)2 = 32
Hence the center is (3, –6),
(3, –3),
and radius is 3.
The Completing-the-Square
method is the basic method for (0, –6), (6, –6),
(3, –6),
handling 2nd degree problems.
(–9, –6)

88. Circles
Example E. Use completing the square to find the center
and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2 – 6x + + y2 + 12y + = –36 complete the squares
x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36
( x – 3 )2 + (y + 6)2 = 9
( x – 3 )2 + (y + 6)2 = 32
Hence the center is (3, –6),
(3, –3),
and radius is 3.
The Completing-the-Square
method is the basic method for (0, –6), (6, –6),
(3, –6),
handling 2nd degree problems.
We summarize the hyperbola
and parabola below.
(–9, –6)

89. Hyperbolas

90. Hyperbolas
Just as all the other conic sections, hyperbolas are defined
by distance relations.

91. Hyperbolas
Just as all the other conic sections, hyperbolas are defined
by distance relations.
Given two fixed points, called foci, a hyperbola is the set
of points whose difference of the distances to the foci is
a constant.

92. Hyperbolas
Just as all the other conic sections, hyperbolas are defined
by distance relations.
Given two fixed points, called foci, a hyperbola is the set
of points whose difference of the distances to the foci is
a constant.
If A, B and C are points on a hyperbola as shown
C
A
B

93. Hyperbolas
Just as all the other conic sections, hyperbolas are defined
by distance relations.
Given two fixed points, called foci, a hyperbola is the set
of points whose difference of the distances to the foci is
a constant.
If A, B and C are points on a hyperbola as shown then
a 1 – a2
C
A
a1
a2
B

94. Hyperbolas
Just as all the other conic sections, hyperbolas are defined
by distance relations.
Given two fixed points, called foci, a hyperbola is the set
of points whose difference of the distances to the foci is
a constant.
If A, B and C are points on a hyperbola as shown then
a1 – a2 = b1 – b2
C
A
a1
a2
b2
B
b1

95. Hyperbolas
Just as all the other conic sections, hyperbolas are defined
by distance relations.
Given two fixed points, called foci, a hyperbola is the set
of points whose difference of the distances to the foci is
a constant.
If A, B and C are points on a hyperbola as shown then
a1 – a2 = b1 – b2 = c2 – c1 = constant.
C
c2 A
a1
c1 a2
b2
B
b1

96. Parabolas
Finally, we illustrate the definition that’s based on distance
measurements of the parabolas.
Given a fixed point F, and a line L, the points that are of equal
distance from F the line L is a parabola.
Hence a = A, b = B, c = C as shown below.
For more information, see:
http://en.wikipedia.org/wiki/Parabola
F
L

97. Parabolas
Finally, we illustrate the definition that’s based on distance
measurements of the parabolas.
Given a fixed point F, and a line L, the points that are of equal
distance from F the line L is a parabola.
Hence a = A, b = B, c = C as shown below.
For more information, see:
http://en.wikipedia.org/wiki/Parabola
P1
F a
A
L

98. Parabolas
Finally, we illustrate the definition that’s based on distance
measurements of the parabolas.
Given a fixed point F, and a line L, the points that are of equal
distance from F the line L is a parabola.
Hence a = A, b = B, c = C as shown below.
For more information, see:
http://en.wikipedia.org/wiki/Parabola
P1
F a
b P2
A
B
L

99. Parabolas
Finally, we illustrate the definition that’s based on distance
measurements of the parabolas.
Given a fixed point F, and a line L, the points that are of equal
distance from F the line L is a parabola.
Hence a = A, b = B, c = C as shown below.
For more information, see:
http://en.wikipedia.org/wiki/Parabola
a
b
c A
B
C