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Asher
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Tyro
Asked: 2021-04-15T04:33:31+00:00 In: ,

Write the equation of an ellipse with foci at ( -6, -3),(0, -3) and a major axis of length 10.

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Algebra

 

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  2. acedstud

    Given that the foci are (-6, -3) and (0, -3), the center of the ellipse is at the midpoint of the foci, thus the center is at \left(\dfrac{-6+0}{2}, \dfrac{-3-3}{2}\right)=(-3, -3). The distance between the center and a focus is called c, thus here, c=3. That is the distance from either (-6, -3) or (0, -3) to (-3, -3) is 3.

    Also, since the length of the major axis is 10 \Rightarrow2a=10\Rightarrow a=\dfrac{10}{2}=5.

    Notice that the foci are along the x-axis, so the major axis is the x-axis and hence, the equation of the ellipse is given by \dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1, where (h, k) is the center of the ellipse and c^2=a^2-b^2.

    Thus, 3^2=5^2-b^2\Rightarrow9=25-b^2\Rightarrow b^2=25-9=16\Rightarrow b=\sqrt{16}=4.

    Therefore, the equation of the ellipse is \dfrac{(x-(-3))^2}{5^2}+\dfrac{(y-(-3))^2}{4^2}=1\Rightarrow\dfrac{(x+3)^2}{25}+\dfrac{(y+3)^2}{16}=1.

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