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Thread: analytic geometry about ellipses

  1. #1
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    analytic geometry about ellipses

    Give the equation of the ellipse having foci at (-1, -1) and (-1, 7) and the length of the semi - minor axis is 8 units.
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  2. #2
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    Hello fishlord40
    Quote Originally Posted by fishlord40 View Post
    Give the equation of the ellipse having foci at (-1, -1) and (-1, 7) and the length of the semi - minor axis is 8 units.
    The centre of the ellipse is at the mid-point of the line joining the foci. So that's at $\displaystyle (-1, 3)$. This means the equation will be of the form:
    $\displaystyle \frac{(x+1)^2}{a^2}+\frac{(y-3)^2}{b^2}=1$
    The axis is along the line $\displaystyle x = -1$, so the major axis is vertical, having foci where $\displaystyle y = 3 \pm be$, $\displaystyle b$ being the length of the semi-major axis and $\displaystyle e$ the eccentricity. This gives:
    $\displaystyle be = 4$
    The length of the semi-minor axis is the value of $\displaystyle a$. So $\displaystyle a = 8$.

    Finally, with an ellipse with a vertical major axis, we have:
    $\displaystyle e^2 = 1-\frac{a^2}{b^2}$
    Eliminate $\displaystyle e$ to find $\displaystyle b$, and hence the equation of the ellipse.

    Grandad
    Last edited by Grandad; Feb 15th 2010 at 12:54 AM. Reason: Correction - I didn't read the question properly!
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