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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) - 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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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.