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Question
If the eccentricity of an ellipse is `5/8` and the distance between its foci is 10, then find latus rectum of the ellipse.
Solution
Equation of an ellipse is `x^2/a^2 + y^2/b^2` = 1
Eccentricity, e = `5/8`, foci = (± ae, 0)
Distance between its foci = ae + ae = 2ae
∴ 2ae = 10
⇒ ae = 5
⇒ `a xx 5/8` = 5
⇒ a = 8
Now b2 = a2(1 – e2)
⇒ b2 = `64(1 - 25/64)`
⇒ b2 = `64 xx 39/64`
⇒ b2 = 39
So, the length of the latus rectum = `(2b^2)/a = (2 xx 39)/8 = 39/4`
Hence, the length of the latus rectum = `39/4`.
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