The center of an ellipse is (-4,0) one vertex is (-8,0) and one co-vertex is (-4,1). Write the equation of the ellipse

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The center of an ellipse is (-4,0) one vertex is (-8,0) and one co-vertex is (-4,1). Write the equation of the ellipse

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Thu Hương 4 years 2021-08-10T03:22:36+00:00 1 Answers 5 views 0

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  1. dangkhoa
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    2021-08-10T03:23:42+00:00
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    Answer:

    The equation of the ellipse is;

    \dfrac{(x + 4)^2}{4^2} + \dfrac{(y - 0)^2}{1^2} = 1

    Step-by-step explanation:

    The parameters of the ellipse are;

    The location of the center of the ellipse = (-4 0)

    The coordinates of the vertex of the ellipse = (-8, 0)

    The coordinates of the the co-vertex = (-4, 1)

    The general form of the equation of an ellipse is presented as follows;

    \dfrac{(x - h)^2}{a^2} + \dfrac{(y - k)^2}{b^2} = 1

    The center of the above equation of an ellipse = (h, k)

    The vertex of the above equation are; (h – a, k) and (h + a, k)

    The co vertex of the above equation are; (h, k – b) and (h, k + b)

    By comparison, we have;

    h = -4, k = 0

    For the vertex, we have;

    When

    h – a = -8

    ∴ -4 – a = -8

    -a = -8 + 4 = -4

    a = 4

    When

    h + a = -8

    -4 + a = -8

    ∴ a = -4

    a = 4 or -4

    We note that a² = 4² = (-4)²

    For the covertex, we have;

    When

    k – b = 1

    0 – b = 1

    b = -1

    When k + b = 1

    0 + b = 1

    b = 1

    ∴ b = 1 or -1

    b² = 1² = (-1)²

    We can therefore write the equation of the ellipse as follows;

    \dfrac{(x - (-4))^2}{4^2} + \dfrac{(y - 0)^2}{1^2} = \dfrac{(x + 4)^2}{4^2} + \dfrac{(y - 0)^2}{1^2} = 1

    Therefore;

    \dfrac{(x + 4)^2}{4^2} + \dfrac{(y - 0)^2}{1^2} = 1

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