Ellipses Date: ____________
Ellipses Standard Equation of an Ellipse Center at (0,0) x2 a2 y2 b2 + = 1 (a, 0) x O (0, –b)
Horizontal Major Axis Vertical Major Axis Co-Vertices Vertices Co-Vertices Vertices
Graph the ellipse. Find the vertices and co-vertices. x2 25 y2 9 + = 1 x y a2 = 25 b2 = 9 a = ±5 b = ±3 (0, 3) (–5, 0) (5, 0) Horizontal Major Axis (0,-3) Vertices: (–5, 0) and (5,0) Co-vertices:(0, 3) and (0,-3)
Graph the ellipse. Find the vertices and co-vertices. x2 9 y2 25 + = 1 x y a2 = 9 b2 = 25 (0, 5) a = ±3 b = ±5 (–3, 0) (3, 0) Vertical Major Axis Vertices: (0,5) and (0,-5) (0,-5) Co-vertices: (-3,0) and (3,0)
Translated Ellipses Standard Equation of an Ellipse Center at (h,k) (x – h)2 a2 (y – k)2 b2 + = 1 y (0, k+b) (h+a, 0) (h,k) (0, k–b) (h–a, 0) x
Graph the ellipse Center = (2,-5) (x – 2)2 36 (y + 5)2 16 + = 1 b2 = 16 a = ±6 b = ±4 Horizontal Major Axis Vertices: (8,-5) and (-4,-5) Co-vertices: (2,-1) and (2,-9)
Graph the ellipse Center = (-3,-1) (x + 3)2 25 (y + 1)2 81 + = 1 b2 = 81 a = ±5 b = ±9 Vertical Major Axis Vertices: (-3,8) and (-3,-10) Co-vertices: (-8,-1) and (2,-1)
Write the equation of the ellipse in standard form. Graph Write the equation of the ellipse in standard form. Graph. Find the center, vertices, and co-vertices. 4x2 + 25y2 = 100 x y 100 x2 25 y2 4 + = 1 Center = (0,0) a2 = 25 b2 = 4 a = ±5 b = ±2 Vertices: (-5,0) and (5,0) Co-vertices: (0,2) and (0,-2)
Write the equation of the ellipse in standard form. Graph Write the equation of the ellipse in standard form. Graph. Find the center, vertices, and co-vertices. 25x2 + 3y2 = 75 75 x y x2 3 y2 25 + = 1 Center = (0,0) a2 = 3 b2 = 25 a ≈ ±1.73 b = ±5 Vertices: (0,5) and (0,-5) Co-vertices: (-1.73,0) and (1.73,0)
x2 – 4x + ____ + 9(y2 + 6y + ___) = -49 +4 +81 Write the equation of the ellipse in standard form. Graph. Find the center, vertices, and co-vertices. x2 + 9y2 – 4x + 54y + 49 = 0 x2 – 4x + 9y2 + 54y = -49 4 9 x2 – 4x + ____ + 9(y2 + 6y + ___) = -49 +4 +81 (x – 2)2 + 9(y + 3)2 = 36 36 (x – 2)2 36 (y + 3)2 4 + = 1
Center = (2,-3) (x – 2)2 36 (y + 3)2 4 + = 1 a2 = 36 b2 = 4 a = ±6 Vertices: (-4,-3) and (8,-3) Co-vertices: (2,-1) and (2,-5)
4(x2 + 6x + ____) + y2 – 4y + ___ = -36 9 4 +36 + 4 Write the equation of the ellipse in standard form. Graph. Find the center, vertices, and co-vertices. 4x2 + y2 +24x – 4y + 36 = 0 4x2 + 24x + y2 – 4y = -36 4(x2 + 6x + ____) + y2 – 4y + ___ = -36 9 4 +36 + 4 4(x + 3)2 + (y – 2)2 = 4 4 (x + 3)2 1 (y – 2)2 4 + = 1
(x + 3)2 1 (y – 2)2 4 = 1 Center = (-3,2) + a2 = 1 b2 = 4 a = ±1 Vertices: (-3,4) and (-3,0) Co-vertices: (-4,2) and (-2,2)
4(x2 – 4x + ____) + 9(y2 + 2y + ___) =11 4 1 +16 +9 Write the equation of the ellipse in standard form. Graph. Find the center, vertices, and co-vertices. 4x2 + 9y2 – 16x +18y – 11 = 0 4x2 – 16x + 9y2 + 18y = 11 4(x2 – 4x + ____) + 9(y2 + 2y + ___) =11 4 1 +16 +9 4(x – 2)2 + 9(y + 1)2 = 36 36 (x – 2)2 9 (y + 1)2 4 + = 1
(x – 2)2 9 (y + 1)2 4 = 1 Center = (2,-1) + a2 = 9 b2 = 4 a = ±3 Vertices: (5,-1) and (-1,-1) Co-vertices: (2,1) and (2,-3)
9.4 Ellipses An ellipse is the set of all points P in a plane such that the sum of the distances from P to two fixed points, F1 and F2, called the foci, is a constant. P P P F1 F2 2a F1P + F2P = 2a
Horizontal Major Axis: 9.4 Ellipses y x2 a2 y2 b2 + = 1 (0, b) (a, 0) (–a, 0) a2 > b2 a2 – b2 = c2 x O F1(–c, 0) F2 (c, 0) (0, –b) length of major axis: 2a length of minor axis: 2b
length of major axis: 2b length of minor axis: 2a 9.4 Ellipses Vertical Major Axis: y x2 a2 y2 b2 + = 1 (0, b) F1 (0, c) (–a, 0) (a, 0) b2 > a2 b2 – a2 = c2 x O F2(0, –c) length of major axis: 2b length of minor axis: 2a (0, –b)
Find the foci. x2 25 y2 9 + = 1 25 – 9 = c2 16 = c2 ±4 = c y (–4, 0) x (4, 0)
Find the foci. x2 9 y2 25 + = 1 25 – 9 = c2 16 = c2 ±4 = c y (0,4) x (0,-4)
Find the foci. x2 100 y2 36 + = 1 100 – 36 = c2 64 = c2 ±8 = c y (–8, 0) (8, 0)
Find the foci. (x – 4)2 16 (y – 3)2 25 + = 1 25 – 16 = c2 9 = c2 (4, 6) 9 = c2 ±3 = c (4, 0)
Find the foci. (x + 1)2 4 (y + 2)2 16 + = 1 16 – 4 = c2 12 = c2 (-1,1.5) ±3.5 ≈ c (-1,-5.5)