Section 10.3
Ellipses and Circles
1st Definition of a Circle
A circle is a conic section formed by a plane
intersecting one cone perpendicular to the axis of
the double-napped cone.
The degenerate conic section that is associated
with a circle is a point.
2nd Definition of a Circle
A circle is the set of all points P in a plane that are
the same distance from a given point.
The given distance is the radius of the circle, and
the given point is the center of the circle.
Standard form of a circle with center C (h, k) and
radius r is
( x  h) 2  ( y  k ) 2  r 2
Example 1
Express in standard form the equation of the
circle centered at (-2, 3) with radius 5.
( x  h)  ( y  k )  r
2
2
2
( x  2) 2  ( y  3) 2  52
( x  2)  ( y  3)  25
2
2
Example 2
Express in standard form the equation of the
circle with center at the origin and radius of 4.
Sketch the graph.
( x  h)  ( y  k )  r
2
2
2
( x  0)  ( y  0)  4
2
2
x  y  16
2
2
2
y




x












Example 3
Find the center and radius of the circle with the
equation
2
2
( x  5)  ( y  1)  32
Center:  5,  1
Radius = 4 2
Example 4
Write the equation for each circle described
below.
a.
The circle has its center at (8, -9) and
passes through the point at (4, -6).
( x  h) 2  ( y  k ) 2  r 2
2
2
2
(4  8)  (6  9)  r
2
16  9  r
2
25  r
( x  8)  ( y  9)  25
2
2
b.
The endpoints of a diameter are at (1, 8) and
(1, -4).
 11 8  4 
center : 
,
  1, 2 
2 
 2
(1  1) 2  (8  2) 2  r 2
0  36  r
2
( x  1) 2  ( y  2) 2  36
End of 1st Day
1st Definition of an Ellipse
An ellipse is a conic section formed by a plane
intersecting one cone not perpendicular to the axis
of the double-napped cone.
The degenerate conic section that is associated
with an ellipse is also a point.
2nd Definition
An ellipse is the set of all points (x, y) in a plane,
the sum of whose distances from two distinct
fixed points (foci) is constant.
d1 + d2 = constant
d1
d2
The line through the foci intersects the ellipse
at two points, called vertices. The chord joining
the vertices is the major axis, and its midpoint is
the center of the ellipse. The chord perpendicular
to the major axis at the center is the minor axis
of the ellipse.
minor axis
major axis
vertex
center
vertex
General Equation of an Ellipse
Ax2 + Cy2 + Dx + Ey + F = 0
If A = C, then the ellipse is a circle.
Standard Equation of an Ellipse
The standard form of the equation of an ellipse,
with center (h, k) and major and minor axes of
lengths 2a and 2b respectively, where 0 < b < a,
 x  h
2
2
y k


2
1
2
a
b
where the major axis is horizontal.
 x  h
2
2
y k


2
2
1
b
a
where the major axis is vertical.
The foci lie on the major axis, c units from the
center, with c2 = a2 – b2.
The eccentricity of an ellipse is
c
e
a
Example 1
Find the center, vertices, the endpoints of the
minor axis, foci, eccentricity, and graph for the
ellipses given in standard form.
x2 y 2

1
81 16
a= 9
b= 4
c = 92  42  65
center:  0, 0 
vertices:  9, 0 
endpoints of the minor axis: 0,  4 

foci:  65, 0

65
eccentricity:
9
y




F2
V1 F1



C









x
V2
Example 2
For the following ellipse, find the center, a, b, c,
vertices, the endpoints of the minor axis, foci,
eccentricity, and graph.
16x2 + y2 − 64x + 2y + 49 = 0
What must you do to the general equation above
to do this example?
Complete the square twice.
16x2 + y2 − 64x + 2y + 49 = 0
16x2 − 64x + y2 + 2y = −49
16  x 2  4 x  ___   y 2  2 y  ___  49  ___  ___
16  x  4 x  4   y  2 y  1  49  64  1
2
2
16  x  2    y  1  16
2
 x  2
1
2
2
y  1


16
2
1
a= 4 b=1 c
16  1  15
What type of ellipse is this ellipse? vertical ellipse?
center: (2, −1)
vertices: (2, 3), (2, −5)
endpoints of the minor axis: (3, −1), (1, −1)


foci: 2, 1  15 , 2,  1  15
15
eccentricity:
4

y


V1
F1
x






C
F2
V2

Example 3
Write the equation of each ellipse in standard
form.
A. Endpoints of the major axis are at (0, ±10)
and whose foci are at (0, ±8).
center: (0, 0)
2
2
x
y
vertical ellipse

1
36 100
a = 10; c = 8
b= 6
B.
The endpoints of the major axis are at (10, 2)
and (–8, 2). The foci are at (6, 2) and (–4, 2).
 10  8 2  2 
center : 
,
  1, 2 
2 
 2
horizontal ellipse
a  10  1  9 c  6  1  5
b 2  81  25  56
 x  1
81
2
y  2


56
2
1
C.
The major axis is 20 units in length and
parallel to the y-axis. The minor axis is 6
units in length. The center is located at
(4, 2).
vertical ellipse
2a  20, a  10 2b  6, b  3
 x  4
9
2
y  2


100
2
1