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प्रश्न
Find the eccentricity of an ellipse, if the length of its latus rectum is one-third of its minor axis.
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उत्तर
Let the equation of the ellipse be
`x^2/"a"^2 + y^2/"b"^2` = 1
It is given that,
l(LR) = `1/3l("minor axis")`
∴ `(2"b"^2)/"a" = 1/3(2"b")`
∴ 3b = a
∴ 9b2 = a2
∴ 9a2(1 – e2) = a2
∴ 9(1 – e2) = 1
∴ 9 – 9e2 = 1
∴ 8 = 9e2
∴ e2 = `8/9`
∴ e = `(2sqrt(2))/3` ... [∵ 0 < e < 1]
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