Question

If $a , b$ are co-prime numbers and satisfying $(2+\sqrt{3})^{\frac{1}{\log _{ a }(2-\sqrt{3})}+\frac{1}{\log _{ b }\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)}}=\frac{1}{12}$, then $(a+b)$ can be is equal to

• A. 13
• B. 5
• C. 7
• D. 8

Answer 1

Answer: (C)

As, $ \frac{1}{\log _a(2-\sqrt{3})}+\frac{1}{\log _b\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)}=\log _{2-\sqrt{3}} a+\log _{\frac{\sqrt{3}-1}{\sqrt{3}+1}} b$
$=\log _{2-\sqrt{3}} a+\log _{2-\sqrt{3}} b =\log _{2-\sqrt{3}} \text { (ab) }$
Now, $(2+\sqrt{3})^{\log _{2-\sqrt{3}}( ab )}=\frac{1}{12} \Rightarrow(2-\sqrt{3})^{\log _{2-\sqrt{3}}\left(\frac{1}{ ab }\right)}=\frac{1}{12}$
$\Rightarrow \frac{1}{ ab }=\frac{1}{12} \Rightarrow ab =12$
As $a , b$ are co-prime numbers, so either $a =4, b =3$ or $a =3, b =4$. Hence, $(a+b)=7$.