If x = ( √(2a + 1) + √(2a − 1) ) / ( √(2a + 1) − √(2a − 1) ), then which of the following is true?(a) x^2 + 2ax + 1 = 0 (b) x^2 − 2ax + 1 = 0 (c) x^2 + 4ax − 1 = 0 (d) x^2 − 4ax + 1 = 0

Given, Hence, option 4 is the correct option.

Mathematics

If x = $\dfrac{\sqrt{2a+1} + \sqrt{2a-1}}{\sqrt{2a+1} - \sqrt{2a-1}}$ , then which of the following is true?
x² + 2ax + 1 = 0
x² − 2ax + 1 = 0
x² + 4ax − 1 = 0
x² − 4ax + 1 = 0

Ratio Proportion

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Answer

Given,

$x = \dfrac{\sqrt{2a+1} + \sqrt{2a-1}}{\sqrt{2a+1} - \sqrt{2a-1}} \\[1em] x\Big(\sqrt{2a+1} - \sqrt{2a-1}\Big) = \sqrt{2a+1} + \sqrt{2a-1} \\[1em] x\sqrt{2a+1} - x\sqrt{2a-1} = \sqrt{2a+1} + \sqrt{2a-1} \\[1em] (x-1)\sqrt{2a+1} = (x+1)\sqrt{2a-1} \\[1em] \text{Squaring on Both Sides,} \\[1em] (x-1)^2(2a+1) = (x+1)^2(2a-1) \\[1em] (2a+1)(x^2 - 2x + 1) = (2a-1)(x^2 + 2x + 1) \\[1em] 2ax^2 - 4ax + 2a + x^2 - 2x + 1 = 2ax^2 + 4ax + 2a - x^2 - 2x - 1 \\[1em] 2ax^2 - 2ax^2 + x^2 + x^2 - 4ax - 4ax + 2a - 2a - 2x + 2x + 1 + 1 = 0\\[1em] 2x^2 - 8ax + 2 = 0 \\[1em] 2(x^2 - 4ax + 1) = 0 \\[1em] x^2 - 4ax + 1 = 0$

Hence, option 4 is the correct option.

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Mathematics

If x = $\dfrac{\sqrt{2a+1} + \sqrt{2a-1}}{\sqrt{2a+1} - \sqrt{2a-1}}$ , then which of the following is true?
x² + 2ax + 1 = 0
x² − 2ax + 1 = 0
x² + 4ax − 1 = 0
x² − 4ax + 1 = 0

Ratio Proportion

Answer

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Answer

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If a = $\dfrac{4xy}{x+y}$ , then $\Big(\dfrac{a+2x}{a-2x} + \dfrac{a+2y}{a-2y}\Big)$ equals :
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