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5 July, 05:04

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?

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  1. 5 July, 05:32
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    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
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