1st Day
Section 10.3
A circle is a set of points in a plane that are agiven distance (radius) from a given point(center).
Standard Form: (x – h)2 + (y – k)2 = r2
Center: (h, k)
Radius: r
Definition of a Circle
Examples
Rewrite in standard form. Then find the center,radius, and graph.
1.x2 + y2 – 4x – 16y + 64 = 0
Complete the square twice.
x2 – 4x + y2 – 16y = -64
x2 – 4x + ___ + y2 – 16y + ____= -64 + ____ + ____
x2 – 4x + 4 + y2 – 16y + 64= -64 + 4 + 64
(x – 2)2 + (y – 8)2 = 4
(x – 2)2 + (y – 8)2 = 4
Center: (2, 8)
Radius: r = 2
2.x2 + y2 – 2x – 2y – 26 = 0
x2 – 2x + y2 – 2y = 26
x2 – 2x + ____ + y2 – 2y + ____ = 26 + ____ + ____
x2 – 2x + 1 + y2 – 2y + 1 = 26 + 1 + 1
(x – 1)2 + (y – 1)2 = 28
(x – 1)2 + (y – 1)2 = 28
Center: (1, 1)
Radius:
Write the equation for a circle with:
3.Center (-3, 2) and radius of 3
(x – h)2 + (y – k)2 = r2
(x – (-3))2 + (y – 2)2 = 32
(x + 3)2 + (y – 2)2 = 9
4.Center (2, -1); goes through (5, 4)
(x – h)2 + (y – k)2 = r2
(5 – 2)2 + (4 – (-1))2 = r2
(3)2 + (5)2 = r2
9 + 25 = r2
34 = r2
(x – 2)2 + (y + 1)2 = 34
2nd Day
An ellipse is a set of points in a plane the sum ofwhose distances from two distinct points (foci) isconstant.
Definition of an Ellipse
VerticalHorizontal
Picture:
Standard Form:
where a > b.
“c” is the distance from the center to a focus.
VerticalHorizontal
Foci:(h, k c)(h c, k)
Major Axis is the segment whose endpoints are thevertices of the ellipse and its length is 2a.
Vertices:(h, k a)(h a, k)
VerticalHorizontal
Minor Axis is the segment perpendicular to themajor axis at the center of the ellipse and its lengthis 2b.
Endpoints of the minor axis:
(h b, k)(h, k b)
Find the missing information and graph.
Type of Ellipse: Horizontal
a2 = 25b2 = 4c2 = 25 – 4 = 21
a = 5b = 2
Example
Center: (-4, 3)
Vertices: (-9, 3); (1, 3)
Endpts. of Minor Axis:
(-4, 5); (-4, 1)
Foci:
Length of Major Axis: 2a = 2(5) = 10
Length of Minor Axis: 2b = 2(2) = 4
3rd Day
Rewrite the equation of the ellipse in standardform and then graph the ellipse.
1.25x2 + 16y2 – 50x – 128y – 119 = 0
Complete the square twice.
25x2 – 50x + 16y2 – 128y = 119
Factor out the coefficient of the squared terms.
25(x2 – 2x) + 16(y2 – 8y) = 119
25(x2 – 2x + __) + 16(y2 – 8y + __) = 119 + __ + __
25(x2 – 2x + 1) + 16(y2 – 8y + 16) = 119 + 25 + 256
25(x – 1)2 + 16(y – 4)2 = 400
Write an equation for each ellipse described.
2.Length of major axis = 14; Foci (4, 0), (-4, 0)
Horizontal ellipse
Center: (0, 0)
c = 4
a = 7
b2 = 49 – 16 = 33
3.Vertices: (2, 8); (2, 0) and minor axisendpoints: (5, 4); (-1, 4)
Vertical ellipse
Center: (2, 4)
a = 4
b = 3
