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If `X = (Sqrt(M + N) + Sqrt(M - N))/(Sqrt(M + N) - Sqrt(M - N))` Express N in Terms of X and M - ICSE Class 10 - Mathematics

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Question

If `x = (sqrt(m + n) + sqrt(m - n))/(sqrt(m + n) - sqrt(m - n))` express n in terms of x and m

Solution

`x = (sqrt(m + n) + sqrt(m - n))/(sqrt(m + n) - sqrt(m - n)`

Applying componendo and dividendo

`(x + 1)/(x - 1) = (sqrt(m + n) + sqrt(m - n) + sqrt(m + n) - sqrt(m - n))/(sqrt(m + n) + sqrt(m - n) - sqrt(m + n) + sqrt(m - n))`

`(x + 1)/(x - 1) =  (2sqrt(m + n))/(2sqrt(m - n))`

Squaring both sides,

`(x^2 + 2x + 1)/(x^2 - 2x + 1) = (m + n)/(m - n)`

Applying componendo and dividendo

`(x^2 + 2x + 1 + x^2 - 2x + 1 )/(x^2 + 2x  + 1 - x^2 + 2x - 1) = 
(m + n + m - n)/(m + n - m + n)`

`(2x^2 + 2)/(4x) = (2m)/(2n)`

`(x^2 + 1)/(2x) = m/n`

`(x^2 + 1)/(2mx) = 1/n`

`n = (2mx)/(x^2 + 1)`

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APPEARS IN

 Selina Solution for Selina ICSE Concise Mathematics for Class 10 (2018-2019) (2017 to Current)
Chapter 7: Ratio and Proportion (Including Properties and Uses)
Ex.7C | Q: 14

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Solution If `X = (Sqrt(M + N) + Sqrt(M - N))/(Sqrt(M + N) - Sqrt(M - N))` Express N in Terms of X and M Concept: Componendo and Dividendo Properties.
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