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If a = (4sqrt6)/(Sqrt2 + Sqrt3) Find the Value of (A + 2sqrt2)/(A - 2sqrt2) + (A + 2sqrt3)/(A - 2sqrt3) - Mathematics

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Question

If a = `(4sqrt6)/(sqrt2 + sqrt3)` find the value of `(a + 2sqrt2)/(a - 2sqrt2) + (a + 2sqrt3)/(a - 2sqrt3)`

Solution

a = `(4sqrt6)/(sqrt2 + sqrt3)`

`a/(2sqrt2) = (2sqrt3)/(sqrt2 + sqrt3)`

Applying componendo and divendo

`(a + 2sqrt2)/(a - 2sqrt2) = (2sqrt3 + sqrt2 + sqrt3)/(2sqrt3 - sqrt2 - sqrt3)`

`(a + 2sqrt2)/(a - 2sqrt2) = (3sqrt3 + sqrt2)/(sqrt3 - sqrt2)`  ....(1)

`a/(2sqrt3) = (2sqrt2)/(sqrt2 + sqrt3)`

Applying componendo and dividendo

`(a + 2sqrt3)/(a - 2sqrt3) = (2sqrt2 + sqrt2 + sqrt3)/(2sqrt2 - sqrt2 - sqrt3)`

`(a + 2sqrt3)/(a - 2sqrt3) = (3sqrt2 + sqrt3)/(sqrt2 - sqrt3)`  ...(2)

From (1) and (2)

`(a + 2sqrt2)/(a - 2sqrt2) + (a + 2sqrt3)/(a - 2sqrt3) = (3sqrt3 + sqrt2)/(sqrt3 - sqrt2) + (3sqrt2 + sqrt3)/(sqrt2 - sqrt3)`

`(a + 2sqrt2)/(a - 2sqrt2) + (a + 2sqrt3)/(a - 2sqrt3) = (3sqrt2 + sqrt3 - 3sqrt3 - sqrt2)/(sqrt2 - sqrt3)`

`(a + 2sqrt2)/(a - 2sqrt2) +(a + 2sqrt3)/(a - 2sqrt3) = (2sqrt2 - 2sqrt3)/(sqrt2 - sqrt3)  = 2`

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APPEARS IN

 Selina Solution for Concise Mathematics for Class 10 ICSE (2020 (Latest))
Chapter 7: Ratio and Proportion (Including Properties and Uses)
Exercise 7(C) | Q: 6.2 | Page no. 101

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Solution If a = (4sqrt6)/(Sqrt2 + Sqrt3) Find the Value of (A + 2sqrt2)/(A - 2sqrt2) + (A + 2sqrt3)/(A - 2sqrt3) Concept: Componendo and Dividendo Properties.
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