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Latus Rectum of Ellipse


It is a chord of the ellipse passing through a focus and perpendicular to the major axis.
Let LL' be the latus rectum of the ellipse
                                       (1)
passing through the focus F2 (ae, 0).
The coordinates of L are (ae, F2L).
Since L lies on (1)
or
Hence, length of the lactus rectum is .

Example
  1. Find the eccentricity of the ellipse
    1. If its latus rectum is half of its minor-axis,
    2. If its latus rectum is half of its major axis.
Solution
  1. Let the equation of the ellipse be
    1. Its latus rectum = and its minor axis = 2b
      Given latus rectum = (minor axis)
      or a = 2b
      Thus,
    2. Latus rectum = and major axis = 2a.
      Given latus rectum = (major axis)

      2b2 = a2 or b2 =
      Thus,  
  2. Find an equation of the ellipse referred to its axes as the coordinate axes with latus rectum of length 4 and distance between foci .
Solution
Let the equation of the ellipse be
Latus rectum = = 4 or b2 = 2a   -------------------(1)
Distance between the foci   ---------------(2)
Since b2 = a2 (1 e2) 2a = a2 a2 e2 = a2 8 [using (2)]
a2 2a - 8 = 0 or (a 4) (a + 2) = 0 a = 4, 2
But a cannot be negative, therefore, a = 4 b2 = 2a = 8
Hence, equation of the ellipse is

 

  1. Find the eccentricity of the ellipse
    1. if the distance between the foci is equal to the length of the latus rectum.
    2. if the minor axis is equal to the distance between the foci.

Solution

  1. Distance between the foci = length of the latus rectum


    Since for an ellipse, 0<, e<1, we reject
  2. Minor axis = distance between the foci 2b = 2ae or b2 = a2 e2
    or a2(1 e2) = a2 e 2 or 1 e2 = e2 or 2e2 = 1 .
  1. Find an equation of the ellipse whose axes are long the coordinate axes, centre at the origin, latus rectum = 8 and   .
Solution
Latus rectum = = 8 b2 = 4a.
Since , from the relation b2 = a2 (1 e2), we get or or a = 8.
Thus, b2 = (4)(8) = 32.
Hence, equation of the required ellipse is
  1. Find the equation of the ellipse whose minor axis is equal to the distance between the foci and whose latus rectum is 10.
Solution
We are given 2b = 2ae or b = ae
b2 = a2 e2 a2 (1 e2) = a2 e2
1 e2 = e2 or 2e2 = 1 or .
Also, since the latus rectum = = 10, we get b2 = 5a a2 (1 e2) = 5a
or a = 10. Thus, b2 = 5 (10) = 50.
Hence, equation of the required ellipse is
  1. Find an equation of the set of all points whose distance from (0,4) is of their distance from the line y = 9.
Solution
Let P(x, y) be any point on the ellipse. By definition distance between P and (0, 4) is times of its distance from the line y = 9.

9[x2 + y2 8y + 16] = 4(y2 18y + 81)
9x2 + 5y2 = 180.




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