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Componendo and Dividendo

Solve the fol...

Question

Solve the following equations:
$\frac{a + 2 x + \sqrt{a^{2} - 4 x^{2}}}{a + 2 x - \sqrt{a^{2} - 4 x^{2}}} = \frac{5 x}{a}$ .

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Solution

$\frac{a + 2 x + \sqrt{a^{2} - 4 x^{2}}}{a + 2 x - \sqrt{a^{2} - 4 x^{2}}} = \frac{5 x}{a}$

$\frac{a + 2 x + \sqrt{a + 2 x} \sqrt{a - 2 x}}{a + 2 x - \sqrt{a + 2 x} \sqrt{a - 2 x}} = \frac{5 x}{a}$

$\frac{\sqrt{a + 2 x} + \sqrt{a - 2 x}}{\sqrt{a + 2 x} - \sqrt{a - 2 x}} = \frac{5 x}{a}$

Applying Componendo and Dividendo

$\frac{\sqrt{a + 2 x}}{\sqrt{a - 2 x}} = \frac{5 x + a}{5 x - a}$

Squaring both sides

$\frac{a + 2 x}{a - 2 x} = \frac{25 x^{2} + a^{2} + 10 a x}{25 x^{2} + a^{2} - 10 a x}$

$\frac{a + 2 x}{a - 2 x} = \frac{(25 x^{2} + a^{2}) + 10 a x}{(25 x^{2} + a^{2}) - 10 a x}$

Again applying Componendo and Dividendo

$\frac{a}{2 x} = \frac{25 x^{2} + a^{2}}{10 a x}$
$\frac{a}{2} = \frac{25 x^{2} + a^{2}}{10 a} (x \neq 0)$
$10 a^{2} = 50 x^{2} + 2 a^{2}$
$50 x^{2} = 8 a^{2} x^{2} = \frac{4 a^{2}}{25}$
$\Rightarrow x = \pm \frac{2 a}{5}$

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