An ellipse has vertices along the major axis at (0, 8) and (0, - 2). The foci of the ellipse are located at (0, 7) and

(0, - 1). What are the values of a, b, h, and k, given the equation below?

The values are a = 5, b = 3, h = 0, k = 3

The equation is x²/9 + (y - 3) ²/25 = 1

Step-by-step explanation:

* Lets revise the standard equation of the ellipse

- The standard form of the equation of an ellipse with center (h, k)

and major axis parallel to y-axis is (x - h) ²/b² + (y - k) ²/a² = 1, where

-The length of the major axis is 2a

- The coordinates of the vertices are (h, k ± a)

- The length of the minor axis is 2b

- The coordinates of the co-vertices are (h ± b, k)

- The coordinates of the foci are (h, k ± c), where c² = a² - b²

* Now lets solve the problem

∵ The vertices of the ellipse along the major axis are (0, 8), (0, - 2)

∴ The major axis is the y-axis

∴ The vertices are (h, k + a) and (h, k - a)

∴ h = 0

∴ k + a = 8 ⇒ (1)

∴ k - a = - 2 ⇒ (2)

∵ The foci of it located at (0, 7), (0, - 1)

∵ The coordinates of the foci are (h, k + c) and (h, k - c)

∴ h = 0

∴ k + c = 7 ⇒ (3)

∴ k - c = - 1 ⇒ (4)

- To find k and a add equations (1) and (2)

∴ (k + k) + (a + - a) = (8 + - 2)

∴ 2k = 6 ⇒ divide both sides by 2

∴ k = 3

- Substitute the value of k in equation (1) or (2) to find a

∴ 3 + a = 8 ⇒ subtract 3 from both sides

∴ a = 5

- To find the value of c substitute the value of k in equation (3) or (4)

∴ 3 + c = 7 ⇒ subtract 3 from both sides

∴ c = 4

- To find b use the equation c² = a² - b²

∵ a = 5 and c = 4

∴ (4) ² = (5) ² - a²

∴ 16 = 25 - b² ⇒ subtract 25 from both sides

∴ - 9 = - b² ⇒ multiply both sides by - 1

∴ b² = 9 ⇒ take √ for both sides

∴ b = 3

* The values are a = 5, b = 3, h = 0, k = 3

* The equation is x²/9 + (y - 3) ²/25 = 1