Question

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Please help me in the above attachment ! ! ! !

Please no short answers!

Answer 1

Let the Two Numbers be : P and Q

We know that Geometric Mean of Two Numbers is Square Root of Product of those Two Numbers.

Geometric Mean of P and Q is $\sqrt{P\times Q}$

Given that the Sum of this Two Numbers is 6 times their Geometric Mean.

⇒ $P + Q = 6\sqrt{P\times Q}$

⇒ $\frac{P + Q}{2\sqrt{PQ}} = \frac{3}{1}$

Applying Componendo and Dividendo Theoram on Both sides We get :

⇒ $\frac{P + Q + 2\sqrt{PQ}}{P + Q - 2\sqrt{PQ}} = \frac{3 + 1}{3 - 1}$

⇒ $\frac{(\sqrt{P} + \sqrt{Q})^2}{(\sqrt{P} - \sqrt{Q})^2} = \frac{2}{1}$

⇒ $\frac{\sqrt{P} + \sqrt{Q}}{\sqrt{P} - \sqrt{Q}} = \frac{\sqrt{2}}{1}$

Applying Componendo and Dividendo Theoram on Both sides We get :

⇒ $\frac{\sqrt{P} + \sqrt{Q} + \sqrt{P} - \sqrt{Q}}{\sqrt{P} + \sqrt{Q} - \sqrt{P} + \sqrt{Q}} = \frac{\sqrt{2} + 1}{\sqrt{2} - 1}$

⇒ $\sqrt{\frac{P}{Q}} = \frac{\sqrt{2} + 1}{\sqrt{2} - 1}$

Squaring on Both sides we get :

⇒ $\frac{P}{Q} = \frac{3 + 2\sqrt{2}}{3 - 2\sqrt{2}}$

Answer 2

hey mate.
here's the solution