Write the standard form of the ellipse and determine each of the following

Problem 1:

1.) 144x² + y² = 144

Solution:

standard form = (x² / 1²) + (y² / 144) = 1

(x - 0)² / 1 ² + (y - 0)² / 12² = 1

h = 0 ; k = 0

center (0, 0)

orientation = vertical because a < b

a = 1  ; semi minor axis

b = 12 ; semi major axis

c² = b² - a²

c² = 1² - 12²

c² = 144 - 1

c = √143

vertices = (0, -12), (0, 12)

co vertices = (-1, 0), (1, 0)

foci = (0, - √143), (0, √143)

major axis is parallel to y axis

eccentricity = c / b

eccentricity = √143 / 12

eccentricity  = 0.9965217285917832

Problem 2:

2.) 7x² = 35 - 5y²

Solution:

7x² + 5y² - 35 = 0

(x² - 0) / (√5)² + (y² - 0) / (√7)² = 1

(x²) / (√5)² + (y²) / (√7)² = 1 ; standard form

orientation = vertical because a < b

h = 0 ; k = 0

center (0, 0)

a = √5 semi minor axis

b = √7 semi major axis

c² = b² - a²

c² = (√7)² - (√5)²

c = √(2)

vertices = (0, - √7), (0, √7)

co vertices = (-√5, 0), (√5, 0)

foci = (0, -√2), (0, √2)

major axis is parallel to y axis

eccentricity = c / b

eccentricity = √2 / √7

eccentricity = 0.5345224838248488

Problem 3:

3.) 4(x - 2)² + 3(y - 1)² = 36

4x² - 16x + 3y² - 6y - 17 = 0

(x - 2)² / 9 + (y - 1)² / 12 = 1

(x - 2)² / 3² + (y - 1)² / ((√4)(√3))² = 1

(x - 2)² / 3² + (y - 1)² / (2√3)² = 1 standard form

orientation = vertical because a < b

h = 2

k = 1

center (2, 1)

a = 3

b = 2√3

c² = b² - a²

c² = (2√3)² - 3²

c² = 12 - 9

c = √3

vertices = (2, 1 - 2√3),  (2, 1 + 2√3)

co vertices = (-1, 1 ), (5, 1)

foci = (2, 1 - √3), (2, 1 + √3)

major axis is parallel to y axis

eccentricity = c / b

eccentricity = √3 / 2√3

eccentricity  = 1/2

Problem 4:

4.) 9x² + 25y² - 126x + 100y + 316 = 0

Solution:

9x² - 126x + 25y² + 100y + 316 =0

9x² (x - 14x + 49) + 25(y² + 4y + 4) = -316 + 441 + 100

9(x -7)² + 25(y + 2)² = 225

(x - 7)² / 5² + (y + 2)² / 3³ = 1

(x - 7)² / 5² + (y  + 2)² / 3³ = 1  standard form

h = 7

k = -2

center = (7, -2)

orientation = horizontal because a > b

a = 5

b = 3

c² = a² - b²

c² = 5² - 3²

c² = 25 - 9

c² = 16

c = √16

c = 4

vertices (2, - 2), (12, -2)

co vertices (7, -5), (7, 1)

foci = (3, -2), (11, -2)

eccentricity = c / a

eccentricity = 4 / 5

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