Write an equation in standard form for each ellipse with center (0, 0) and co-vertex at (5, 0); focus at (0, 3).

Question

Write an equation in standard form for each ellipse with center (0, 0) and co-vertex at (5, 0); focus at (0, 3).

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Ivy 1 week 2021-09-15T20:17:41+00:00 1 Answer 0

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    2021-09-15T20:19:22+00:00

    Answer:

    The required standard form of  ellipse is \frac{x^2}{25}+\frac{y^2}{34}=1.

    Step-by-step explanation:

    The standard form of an ellipse is

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

    Where, (h,k) is center of the ellipse.

    It is given that the center of the circle is (0,0), so the standard form of the ellipse is

    \frac{x^2}{a^2}+\frac{y^2}{b^2}=1             …. (1)

    If a>b, then coordinates of vertices are (±a,0), coordinates of co-vertices are (0,±b) and focus (±c,0).

    c^2=a^2-b^2            …. (2)

    If a<b, then coordinates of vertices are (0,±b), coordinates of co-vertices are (±a,0) and focus (0,±c).

    c^2=b^2-a^2            …. (3)

    It is given that co-vertex of the ellipse at (5, 0); focus at (0, 3). So, a<b we get

    a=5,c=3

    Substitute a=5 and c=3 these values in equation (3).

    3^2=b^2-(5)^2

    9=b^2-25

    34=b^2

    \sqrt{34}=b

    Substitute a=5 and b=\sqrt{34} in equation (1), to find the required equation.

    \frac{x^2}{5^2}+\frac{y^2}{(\sqrt{34})^2}=1

    \frac{x^2}{25}+\frac{y^2}{34}=1

    Therefore the required standard form of  ellipse is \frac{x^2}{25}+\frac{y^2}{34}=1.

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