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# If X = (Sqrt(A + 1) + Sqrt(A - 1))/(Sqrt(A + 1) + Sqrt(A - 1)), Using Properties of Proportion Show that X^2 - 2ax + 1 = 0 - Mathematics

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#### Question

If x = (sqrt(a + 1) + sqrt(a - 1))/(sqrt(a + 1) + sqrt(a - 1)), using properties of proportion show that x^2 - 2ax + 1 = 0

#### Solution

Given that x = (sqrt(a + 1) + sqrt(a - 1))/(sqrt(a + 1) - sqrt(a- 1))

By applying Componendo-Dividendo

=> (x + 1)/(x - 1) = ((sqrt(a + 1) + sqrt(a - 1)) + (sqrt(a + 1) - sqrt(a - 1)))/((sqrt(a + 1) + sqrt(a - 1)) - (sqrt(a + 1) - sqrt(a - 1)))

=> (x + 1)/(x - 1) = (2sqrt(a + 1))/(2sqrt(a - 1))

=> (x + 1)/(x - 1)  = sqrt(a + 1)/sqrt(a -1 )

Squaring both the sides of the equation we have

=> ((x + 1)/(x - 1))^2 = (a + 1)/(a - 1)

=> (x + 1)^2 (a - 1) = (x - 1)^2 (a + 1)

=> (x^2 + 2x + 1) - (x^2 + 2x + 1)= a(x^2 - 2x + 1) + (x^2 - 2x + 1)

=> 4ax = 2x^2 + 2

=> 2ax = x^2 + 1

=> x^2 - 2ax + 1 = 0

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If X = (Sqrt(A + 1) + Sqrt(A - 1))/(Sqrt(A + 1) + Sqrt(A - 1)), Using Properties of Proportion Show that X^2 - 2ax + 1 = 0 Concept: Componendo and Dividendo Properties.