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If the value of \(\frac{{3x\sqrt y + 2y\sqrt x }}{{3x\sqrt y - 2y\sqrt x }} - \frac{{3x\sqrt y - 2y\sqrt x }}{{3x\sqrt y + 2y\sqrt x }}\) is same as that of \(\sqrt x\sqrt y\), then which of the following relations between x and y is correct?

If the value of \(\frac{{3x\sqrt y + 2y\sqrt x }}{{3x\sqrt y - 2y\sqrt x }} - \frac{{3x\sqrt y - 2y\sqrt x }}{{3x\sqrt y + 2y\sqrt x }}\) is same as that of \(\sqrt x\sqrt y\), then which of the following relations between x and y is correct? Correct Answer 9x - 4y = 24

Given:

- = √x × √y

Concept Used:

We need to take LCM and simplify the equation.

Formula Used:

(a + b)2 - (a - b)2 = 4ab

(a2 - b2) = (a + b)(a - b)

Calculation:

Take a = 3x√y and b = 2y√x

Given expression will be transformed to -

⇒ / = √(xy)

⇒ 4ab/(a2 - b2) = √(xy)

⇒ (4 ×  3x√y × 2y√x)/ = √(xy)

⇒ (24x√x × y√y)/(9x2y - 4y2x) = √(xy)

⇒ /[(xy)(9x - 4y) = √(xy)

⇒ 24/(9x - 4y) = 1

⇒ 9x - 4y = 24

∴ The correct relation between x and y is 9x - 4y = 24

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