Question : If $(2+\sqrt{3})a=(2-\sqrt{3})b=1$, then the value of $\frac{1}{a}+\frac{1}{b}$ is:
Option 1: $1$
Option 2: $2$
Option 3: $2\sqrt{3}$
Option 4: $4$
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Correct Answer: $4$
Solution :
$(2+\sqrt{3})a=(2-\sqrt{3})b=1$
$(2+\sqrt{3})a=1$
$⇒ a = \frac{1}{2+\sqrt{3}}$
$⇒ \frac{1}{a} = 2-\sqrt{3}$
Again, $(2-\sqrt{3})b=1$
$⇒ b = \frac{1}{2-\sqrt{3}}$
$⇒ \frac{1}{b} = 2+\sqrt{3}$
$\frac{1}{a} + \frac{1}{b}=2-\sqrt{3}+2+\sqrt{3}$
$\frac{1}{a} + \frac{1}{b}=4$
Hence, the correct answer is $4$.
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