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1 April, 10:17

Write an equation of an ellipse in standard form with the center at the origin and a height of 12 units and width of 19 units.

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  1. 1 April, 10:43
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    The equation of the ellipse in standard form is 4x²/361 + y²/36 = 1

    Step-by-step explanation:

    * Lets revise the equation of the ellipse

    - The standard form of the equation of an ellipse with center (0, 0)

    and major axis parallel to the x-axis is x²/a² + y²/b² = 1

    # a > b

    - The length of the major axis is 2a

    - The coordinates of the vertices are (± a, 0)

    - The length of the minor axis is 2b

    - The coordinates of the co-vertices are (0, ± b)

    - The coordinates of the foci are (± c, 0), where c ² = a ² - b²

    * Lets solve the problem

    ∵ The center of the ellipse is (0,0)

    ∵ Its width is 19 units

    ∴ The length of the major axis is = 19

    ∴ 2a = 19 ⇒ divide both sides by 2

    ∴ a = 19/2 ⇒ ∴ a² = 361/4

    ∵ Its height is 12 units

    ∴ The length of the minor axis is = 12

    ∴ 2b = 12 ⇒ divide both sides by 2

    ∴ b = 12/2 = 6 ⇒ ∴ b² = 36

    - Lets write the equation in standard form

    ∵ The equation is x²/a² + y²/b² = 1

    ∴ x² / (361/4) + y²/36 = 1 ⇒ simplify it

    ∴ 4x²/361 + y²/36 = 1

    * The equation of the ellipse in standard form is 4x²/361 + y²/36 = 1
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