Conic Sections
©Mathworld
Introduction
They can be
described or
defined as a set
of points which
satisfy certain
conditions
• We will consider various conic sections and
how they are described analytically
 Circles
 Parabolas
 Hyperbolas
 Ellipses
Conic Sections - Definition
• A conic section is a
curve formed by
intersecting cone with
a plane
• There are four types
of Conic sections
Conic Sections - Four Types
Colleen Beaudoin
February, 2009
At the end of the lesson, the student is able to:
(1) define a circle;
(2) determine the standard form of equation of
a circle;
(3) graph a circle in a rectangular coordinate
system; and
(4) solve situational problems involving conic
sections (circles).
A circle is a set of points a given distance from
one point called the center.
The distance from the center is called the radius


Review: The geometric definition relies on a
cone and a plane intersecting it
Algebraic definition: a set of points in the
plane that are equidistant from a fixed point on
the plane (the center).
Find the distance from the center of the circle (h,k)
to any point on the circle (represented by (x,y)).
This is the radius of the circle.
Review the distance formula:
d  ( x2  x1 )2  ( y2  y1 )2
(x,y)
Substitute in the values.
r  ( x  h) 2  ( y  k ) 2
Square both sides to get
the general form of a
x
circle in center-radius form.
r  ( x  h)  ( y  k )
2
2
r
(h,k)
2
y
Radius (r)
Center (h,k)




Both variables are squared.
Equation of a circle in center-radius form a.k.a
standard form of the equation of the circle.
What makes the circle different from a line?
What makes the circle different from the
parabola?
Equations of a Circle
Standard Equation of a Circle
Definition of a Circle
Standard equation of the circle
If the circle is at the origin
x2+y2=r2
r is the radius
If the circle is not at the origin
 x  h
2
 y  k  r
2
The center is at (h, k)
2
Standard equation of the circle
If the circle is at the origin
x2+y2=r2
r is the radius
If the circle is not at the origin
 x  h
2
 y  k  r
2
Solve for r
r
 x  h
2
 y  k
2
2
Write the equation of Circle
Center at ( -5,0) and radius 4.8
 x  h
2
h =?
k =?
r = 4.8
 y  k  r
2
2
Write the equation of Circle
Center at ( -5,0) and radius 4.8
 x  h
2
h =-5
k =0
r = 4.8
 y  k  r
2
2
 x   5
  y  0  4.8
 x  5
2
2
2
2
 y  23.04
2
Write the equation of Circle
Center at ( 4, -3) and
a point on the circle (2,1)
 x  h
2
h=
k=
r=
x=
y=
 y  k  r
2
2
Write the equation of Circle
Center at ( 4, -3) and
a point on the circle (2,1)
 x  h
2
h=4
k=-3
r=
x=2
y=1
 y  k  r
2
2  4  1   3
 2  4  r
2
2
2
2
4  16  r  20
2
2
r
2
2
Write the equation of Circle
Center at ( 4, -3) and
a point on the circle (2,1)
 x  h
2
h=4
k=-3
r2 = 20
x=2
y=1
 x  4
 x  4
 y  k  r
2
2
2
  y   3  20
2
  y  3  20
2
2
Given the equation of the Circle
(x – 13)2 + (y – 6)2 = 52
Tell if the point is inside or outside the circle.
(5, 6)
Given the equation of the Circle
(x – 13)2 + (y – 6)2 = 52
Tell if the point is inside or outside the circle.
(5, 6)
(5 -13)2 + (6 – 6)2 is it <, >, or = 52
( - 8)2 + 02 = 64
= 25
Greater then outside the circle.
Less then Inside the circle.
Equal is on the circle.
Given the equation of the Circle
(x – 13)2 + (y – 6)2 = 52
Tell if the point is inside or outside the circle.
(5, 6)
Outside
(14,8)
Given the equation of the Circle
(x – 13)2 + (y – 6)2 = 52
Tell if the point is inside or outside the circle.
(5, 6)
Outside
(14,8)
In
(20, 9)
Given the equation of the Circle
(x – 13)2 + (y – 6)2 = 52
Tell if the point is inside or outside the circle.
(5, 6)
Outside
(14,8)
In
(20, 9)
Outside
Given the equation of the Circle
(x – 13)2 + (y – 6)2 = 52
Tell if the point is inside or outside the circle.
(5, 6)
Outside
(14,8)
In
(20, 9)
Outside
(16, 2)
Given the equation of the Circle
(x – 13)2 + (y – 6)2 = 52
Tell if the point is inside or outside the circle.
(5, 6)
Outside
(14,8)
In
(20, 9)
Outside
(16, 2)
On
Examples
Answers
center is (3, 1) r  4
center is (5, 2) r  15
center is (0, 2) r  3
4. Write the equation of a circle centered at (2,-7)
and having a radius of 5.
(x - 2)2 + (y + 7)2 = 25
5. Describe (x - 2)2 + (y + 1)2 = 0
A center point at (2,-1)
6. Describe (x + 1)2 + (y - 3)2 = -1
No graph
7. Write the equation of a circle whose diameter is the
line segment joining A(-3,-4) and B(4,3).
What must you find first?
The center and the radius.
How can you find the center?
The center is the midpoint of the segment.
(½ , - ½ )
How can you find the radius?
The radius is the distance from the center to a
point on the circle. Use the distance formula.
r=
7 2
2
The equation is:
2
2
1 
1
49

x   y  
2 
2
2

General Form of the Equation of the circle
x2 + y2 + Cx + Dy + E = 0
Note: You have to know how to
convert the equation from standard
to general and vice versa.
Write in center-radius form and sketch:
x  y  6 x  4 y  12
2
2
Hint: You must complete the square.
x 2  6 x  ___  y 2  4 y  ___  12
x  6 x  9  y  4 y  4  12  9  4
2
2
( x  3)  ( y  2)  25
2
2
Seatwork
Answer
1. x2 + y2 = 11
2. (x + 6)2 + (y − 7)2 = 36
3. (x − 2)2 + (y − 3)2 = 26
7. Write the standard equation of the circle.
State the center & radius.
x  y  8x  7  0
2
2
( x  8x
)  y  7
2
2
( x  8 x  16)  y  7  16
2
2
( x  4)  y  9
2
2
Center: (4, 0) radius: 3
8. Write the standard equation of the circle.
State the center & radius.
x
x
2
2
 4x
y
2
 6y
3
 4x  4   y  6 y  9  3  4  9
2
2
2
 x  2    y  3
 16
Center: (-2, 3) radius: 4
10. Write the standard equation of the circle.
State the center & radius.
2 x  2 y  16 x  4 y  20  0
2
2
x  y  8 x  2 y  10  0
2
x
x
2
2
2
y
 8x
2
 2y
  10
 8 x  16    y  2 y  1  10  16  1
 x  4
2
2
  y  1  7
2
Center : (4, 1) Radius : 7  2.6
x2 + y2 = 8
center at (15,−20), radius 9
(x − 15)2 + (y + 20)2 = 81
center at (15,−20), radius 9
(x − 15)2 + (y + 20)2 = 81
center at (5, 6), through (9, 4)
(x − 5)2 + (y − 6)2 = 20
center at (−2, 3), tangent to
the x-axis
(x + 2)2 + (y − 3)2 = 9
center at (−2, 3), tangent to
the y-axis
(x + 2)2 + (y − 3)2 = 4
center at (−2, 3), tangent to
the line y = 8
(x + 2)2 + (y − 3)2 = 25
center at (−2, 3), tangent to
the line x = −10
(x + 2)2 + (y − 3)2 = 64
center in the third
quadrant, tangent to both
the x-axis and y-axis,
radius 7
(x + 7)2 + (y + 7)2 = 49
a diameter with endpoints
(−9, 2) and (15, 12)
(x − 3)2 + (y − 7)2 = 169
Write the standard equation of the circle. State
the center & radius and graph the equation.
x  y  8x  7  0
2
2
x  y  4x  6 y  3  0
2
2
2 x  2 y  16 x  4 y  20  0
2
2