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Question
Find the equation of the ellipse in standard form if the distance between directrix is 18 and eccentricity is `1/3`.
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Solution
Let the required equation of ellipse be
`x^2/"a"^2 + y^2/"b"^2` = 1, where a > b.
Given, eccentricity (e) = `1/3`
Distance between directrices = `(2"a")/"e"`
Given, distance between directrices = 18
∴ `(2"a")/"e"` = 18
∴ `(2"a")/(1/3)` = 18
∴ 6a = 18
∴ a = `18/6` = 3
∴ a2 = 9
Now, b2 = a2 (1 – e2)
= `9[1 - (1/3)^2]`
= `9(1 - 1/9)`
= `9(8/9)`
= 8
∴ The required equation of ellipse is `x^2/9 + y^2/8` = 1.
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