Question 446169: A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is 1/2. Then the length of the semi−major axis is
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Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! Without loss of generality, let h > 0, let (h, 0) be the center of the ellipse, and let the point (h + c, 0) be the other focus. Since the (absolute) distance of the center of the ellipse from either foci is c = a*e (where a = semi-major axis, and e = eccentricity), this implies that h = c.
Also, h - a*e = c - a*e = 0 by hypothesis, hence, c = a/2.
Now using the directrix condition, we get h + a/e = 4, or
h + a/(1/2) = h + 2a = 4. After using the fact that h = c, we get c + 2a = 4.
Hence by substitution, a/2 + 2a = 4,
or  , or
 , the length of the semi-major axis.
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