Find the equations of the ellipse, passing through (2,sqrt(30)/3) , distance between directrices 24sqrt(7)/3 `sqrt(7)/3`

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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Since the problem provides the information that the distance between directrices is of `sqrt7/3` , you need to use the following formula, such that:

Distance between directrices `= 2a/e`

`e = (sqrt(a^2 - b^2))/a =>` Distance between directrices = `2a^2/sqrt(a^2 - b^2))`

`sqrt7/3 = 2a^2/sqrt(a^2 - b^2))`

`7/9 = 4a^4/(a^2 - b^2) => 36a^4 = 7(a^2 - b^2)`

The problem also provides the information that the ellipse passes through the point `(2,sqrt(30)/3)` , hence, you need to use the ellipse equation, such that:

`x^2/a^2 + y^2/b^2 = 1 => 4/a^2 + (30/9)/b^2 = 1`

`4b^2 + 10/3a^2 = (ab)^2`

`{(12b^2 + 10a^2 = 3(ab)^2),(-7b^2 + 7a^2 = 36a^4):}`

`{(84b^2 + 70a^2 = 21(ab)^2),(-84b^2 + 84a^2 = 432a^4):}`

`154a^2 = 21(ab)^2 + 432a^4`

`154a^2 - 21(ab)^2 = 432a^4`

`a^2(154 - 21b^2) = 432a^4 => 154 - 21b^2 = 432a^2`

`154 - 432a^2 = 21b^2 => b^2 = (154 - 432a^2)/21`

`36a^4 = 7(a^2 - (154 - 432a^2)/21)`

`756a^4 = 3171a^2 - 1078`

`756a^4 - 3171a^2 + 1078 = 0`

You need to substitute t for `a^2` such that:

`756t^2 - 3171t +1078 = 0`

`t_(1,2) = (3171+-2602))/1512`

`t_1 = 3.81 t_2 = 0.37`

Hence, `a^2 = 3.81` and `a^2 = 0.37` and you may evaluate `b^2` such that:

`b^2 = (154 - 432a^2)/21`

Since `b^2 < 0` for both values of `a^2` , hence, you cannot evaluate the equation of ellipse, under the given conditions provided by the problem.

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