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Question

Find the eccentricity equation of an ellipse whose latus-rectum is
(i) half of its minor axis
(ii) half of its major axis.

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Solution

Let 2a and 2b the minor and major axes of the ellipse.
(i) When latus-rectum is half of minor axis.
2b2a=12×2b
2b2=ab
b2b=a2
b=a2
b2=a24
Now, b2=a2(1e2)
a24=a2(1e2)
14=1e2
e2=114
e2=34
e=32
(ii) When latus-rectum is half of major-axis.
2b2a=12×2a
2b2=a2
a2=2b2
Now,
b2=a2(1e2)
b2=2b2(1e2)
1=2(1e2)
1=22e2
2e2=21
e2=12
e=12

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