Ellipse Analyzing an Ellipse Warmup Graph and find

Ellipse Analyzing an Ellipse

Warm-up Graph and find the Vertices and Foci of the following ellipse 1. 2 13 x + Standar d: 2 17 y = 221 x 2 + y 2 = 1 17 13 Vertices: ( ± √ 17 , 0) ( 0 , ± √ 13 ) Foci: ( ± 2 , 0) 2. 2 16 x + Standar d: 2 9 y = 144 x 2 + y 2 = 1 16 9 Vertices: ( 0 , ± 4 ) (± 2, 0) Foci: (0 , ±√ 7)

Finding the Standard Equation of an Ellipse [C(h, k)] Example: find the standard form of the equation of the ellipse having a foci at (0, 1) and (4, 1) and a major axis of length 6 2 a=major 2 a=6 a=3 To find “a”: axis (0, 1) � (2, 1) � � c=2 Center : (2, 1) h, k c= 2 a= 3 b 2= 5 (4, 1) 2 To find “b”: c = a 2 - b 2 2 2 = 3 2 - b 2 4 = 9 - b 2 = 9 -4 b 2 = 5 (x-2)2 + (y-1)2 = 1 5 9

Analyzing an Ellipse x + 4 y + 6 x - 8 y + 9= 2 2 0 2 (x + 6 x (x 2 + 6 x )+ 2 (4 y - 8 y ) = -9 ) + 4(y 2 - 2 y ) = -9 2 [(½) (6)] 2 [(½) (2)] 4(y 2 - 2 y (x 2 + 6 x +9 ) + +1) = -9 +9 2 2 +1 (4) (x +3 ) + 4(y- 1) = 4 2 2 (x +3 ) + 4(y- 1) = 4 4 (x +3 )2 +(y - 1)2 = 1 4 1

(x +3 )2 +(y - 1)2 = 1 4 1 CENTER: VERTICES: 2 a = 4 a=2 2 b =1 b=1 2 c = 3 c = √ 3 (-3, 1) (-5, 1) (-1, 1) FOCI: (-3+√ 3 , 1) (-3 -√ 3 , 1)

Your turn! 2 4 x 2 (4 x - 8 x 4(x 2 - 2 x 2 [(½) (2)] 2 + 2 y - )+ 2 (y 8 x + 4 y - 8= 0 + 4 y )=8 ) + (y 2 + 4 y )= 8 [(½)2(4)]2 4(x - 2 x +1 ) + (y + 4 y +4) = 8 + 2 2 (4) 1 +4 4(x - 1 ) + (y + 2) = 16 2 2 4(x - 1 ) + (y +2) = 16 16 (x - 1 )2 +(y + 2)2 = 1 4 16

(x - 1 )2 +(y + 2)2 = 1 4 16 2 a = 4 a=2 2 b = 16 b = 4 2 c = 12 c = 2√ 3 CENTER: (1, -2) VERTICES: (1, -6) (1, 2) FOCI: (1, -2+2√ 3 ) (1, -2+√ 3 )

Practice 1. 2 (x-4) = 32 (y+2) The vertex is (4, -2) The focus is (4, 6) The equation of the directrix is y = – 10 The axis of symmetry is x=4 2. Write this equation in standard form: (x - 1 2 2 4 4 x + y - 8 x + 4 y - 8= 0 2 ) 2 2) +(y + 16 =1