Ellipses Unit 7.2

Description Locus of points in a plane such that the sum of the distances from two fixed points, called foci is constant. P Q d 1 + d 2 = d 3 + d 4

Practical importance Chain on a bicycle wheel

Parts of the ellipse Covertex – endpoints of minor axis Vertex – endpoints of major axis Center – midpoint of the major axis Minor Axis – segment through the center and perpendicular to major axis Major Axis – segment that contains the foci Focus – two fixed points

Standard forms of Equations (x – h) 2 + (y – k) 2 = 1 a 2 b 2 (x – h) 2 + (y – k) 2 = 1 b 2 a 2

Equations (x – h) 2 + (y – k) 2 = 1 a 2 b 2 Center: (h, k) Foci: (h ± c, k) Vertices: (h ± a, k) Co-vertices: (h, k± b) Major axis: y = k Minor axis: x = h a, b, c relationship c 2 = a 2 – b 2 x – h) 2 + (y – k) 2 = 1 b 2 a 2 Center: (h, k) Foci: (h, k ± c) Vertices: (h, k ± a) Co-vertices: (h± b, k) Major axis: x = h Minor axis: y = k a, b, c relationship c 2 = a 2 – b 2

Graph Ellipses (x – 3) 2 + (y + 1) 2 = h = 3, k = -1, a = 6, b = 3, c = √36 – 9 = 3√3 Center: (h, k) = 3, -1 Foci: (h ± c, k) = 3 ± 3√3, -1) Vertices: (h ± a, k) = (3 ± 6, -1) =(9, -1) & (-3, -1) Co-vertices: (h, k ± b)=(3,-1 ± 3) = (3, -4) & (3, 2) Major axis: y = k = -1 Minor axis: x = h = 3

Problems Unit 7.2 Problems 1, 2

Completing the Square 4x 2 + y 2 – 24x + 4y + 24 = 0 4x 2 -24x + y 2 + 4y = -24 (group like terms) 4(x 2 - 6x) + y 2 + 4y = -24 (factor) 4(x 2 - 6x + 9) + (y 2 + 4y + 4)= (4) + 4 4(x – 3) 2 + (y+ 2) 2 = 16 (x – 3) 2 + (y+ 2) 2 =

Determine Eccentricity Formula e = c/a Ellipse: 0 < e < 1 Problem 14 (x + 5) 2 + (y – 3) 2 = a = 6√2 c =√ = 18 = 3√2 e = 3√2/ 6√2 =.5

Unit 7.2 Page 438 Problems 1 – 6,