Question : If $\sqrt{\frac{\mathrm{a}}{\mathrm{b}}}=\frac{8}{3}-\sqrt{\frac{\mathrm{b}}{\mathrm{a}}}$ and $a-b=10$, then the value of $ab$ is:
Option 1: $32 \frac{1}{7}$
Option 2: $32 \frac{3}{7}$
Option 3: $32 \frac{4}{7}$
Option 4: $32 \frac{2}{7}$
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Correct Answer: $32 \frac{1}{7}$
Solution :
Given: $\sqrt{\frac{\mathrm{a}}{\mathrm{b}}}=\frac{8}{3}-\sqrt{\frac{\mathrm{b}}{\mathrm{a}}}$
⇒ $\sqrt\frac{a}{b} + \sqrt\frac{b}{a} = \frac{8}{3}$
Squaring both sides, we have,
⇒ $\frac{a}{b}+\frac{b}{a}+2=\frac{64}{9}$
⇒ $\frac{a^2 + b^2}{ab} = \frac{64-18}{9}$
⇒ $a^2 + b^2 = \frac{46ab}{9}$
Now, $a-b=10$
Squaring both sides,
⇒ $a^2 + b^2 - 2ab = 100$
⇒ $\frac{46ab}{9} - 2ab = 100$
⇒ $46ab - 18ab = 900$
⇒ $28ab = 900$
$\therefore ab = \frac{900}{28}=32\frac{1}{7}$
Hence, the correct answer is $32\frac{1}{7}$.
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