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Given X = `(Sqrt(A^2 + B^2) + Sqrt(A^2 - B^2))/(Sqrt(A^2 + B^2) - Sqrt(A^2 - B^2))` Use Componendo and Dividendo to Prove that `B^2 = (2a^2x)/(X^3 + 1)` - ICSE Class 10 - Mathematics

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Question

Given x = `(sqrt(a^2 + b^2) + sqrt(a^2 - b^2))/(sqrt(a^2 + b^2) - sqrt(a^2 - b^2))`

Use componendo and dividendo to prove that `b^2 = (2a^2x)/(x^3 + 1)`

Solution

x = `(sqrt(a^2 + b^2) + sqrt(a^2 - b^2))/(sqrt(a^2 + b^2) - sqrt(a^2 - b^2))`

By componendo and dividendo,

`(x + 1)/(x - 1) = (sqrt(a^2 + b^2) + sqrt(a^2 - b^2) + sqrt(a^2 + b^2) - sqrt(a^2 - b^2))/(sqrt(a^2 + b^2) + sqrt(a^2 - b^2) - sqrt(a^2 + b^2) + sqrt(a^2 - b^2))`

(x + 1)/(x - 1) = `(2sqrt(a^2 + b^2))/(2sqrt(a^2 - b^2))`

Squaring both sides,

`(x^2 + 2x + 1)/(x^2 - 2x + 1) = (a^2 + b^2)/(a^2 - b^2)`

By componendo and dividend

`((x^2 + 2x + 1)+ (x^2 - 2x + 1))/((x^2 + 2x +1) - (x^2 - 2x + 1)) = ((a^2 + b^2) + (a^2 - b^2))/((a^2 + b^2)-(a^2 - b^2))`

`=> (2(x^2 + 1))/"4x" = (2a^2)/(2b^2)`

`=> (x^2 + 1)/(2x) = a^2/b^2`

`=> b^2 = (2a^2x)/(x^2 + 1)`

Hence Proved

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APPEARS IN

 2009-2010 (March) (with solutions)
Question 11.3 | 4.00 marks

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Solution Given X = `(Sqrt(A^2 + B^2) + Sqrt(A^2 - B^2))/(Sqrt(A^2 + B^2) - Sqrt(A^2 - B^2))` Use Componendo and Dividendo to Prove that `B^2 = (2a^2x)/(X^3 + 1)` Concept: Componendo and Dividendo Properties.
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