1 st Day Section 10.3

A circle is a set of points in a plane that are a given distance (radius) from a given point (center). Standard Form: (x – h) 2 + (y – k) 2 = r 2 Center: (h, k) Radius: r Definition of a Circle

Examples

Rewrite in standard form. Then find the center, radius, and graph. 1.x 2 + y 2 – 4x – 16y + 64 = 0 Complete the square twice. x 2 – 4x + y 2 – 16y = -64 x 2 – 4x + ___ + y 2 – 16y + ____= ____ + ____ x 2 – 4x y 2 – 16y + 64= (x – 2) 2 + (y – 8) 2 = 4

Center: (2, 8) Radius: r = 2

2.x 2 + y 2 – 2x – 2y – 26 = 0 x 2 – 2x + y 2 – 2y = 26 x 2 – 2x + ____ + y 2 – 2y + ____ = 26 + ____ + ____ x 2 – 2x y 2 – 2y + 1 = (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 = r 2 (x – (-3)) 2 + (y – 2) 2 = 3 2 (x + 3) 2 + (y – 2) 2 = 9

4.Center (2, -1); goes through (5, 4) (x – h) 2 + (y – k) 2 = r 2 (5 – 2) 2 + (4 – (-1)) 2 = r 2 (3) 2 + (5) 2 = r = r 2 34 = r 2 (x – 2) 2 + (y + 1) 2 = 34

2 nd Day

An ellipse is a set of points in a plane the sum of whose distances from two distinct points (foci) is constant. 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 the vertices of the ellipse and its length is 2a. Vertices:(h, k  a)(h  a, k)

VerticalHorizontal Minor Axis is the segment perpendicular to the major axis at the center of the ellipse and its length is 2b. Endpoints of the minor axis: (h  b, k)(h, k  b)

Find the missing information and graph. Type of Ellipse: Horizontal a 2 = 25b 2 = 4c 2 = 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

3 rd Day

Rewrite the equation of the ellipse in standard form and then graph the ellipse. 1.25x y 2 – 50x – 128y – 119 = 0 Complete the square twice. 25x 2 – 50x + 16y 2 – 128y = 119 Factor out the coefficient of the squared terms. 25(x 2 – 2x) + 16(y 2 – 8y) = (x 2 – 2x + __) + 16(y 2 – 8y + __) = __ + __ 25(x 2 – 2x + 1) + 16(y 2 – 8y + 16) = (x – 1) (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 b 2 = 49 – 16 = 33

3.Vertices: (2, 8); (2, 0) and minor axis endpoints: (5, 4); (-1, 4) Vertical ellipse Center: (2, 4) a = 4 b = 3