If 4 3 + 5 2 48 + 18 = a + b 6 , then the value of a and b are respectively?

Answer: Option D

$$\begin{array}{rl}& \frac{4\sqrt{3}+5\sqrt{2}}{\sqrt{48}+\sqrt{18}}=a+b\sqrt{6}\\ & \Rightarrow \frac{4\sqrt{3}+5\sqrt{2}}{\sqrt{16\times 3}+\sqrt{9\times 2}}=a+b\sqrt{6}\\ & \Rightarrow \frac{4\sqrt{3}+5\sqrt{2}}{4\sqrt{3}+3\sqrt{2}}=a+b\sqrt{6}\\ & \Rightarrow \frac{4\sqrt{3}+5\sqrt{2}}{4\sqrt{3}+3\sqrt{2}}\times \frac{4\sqrt{3}-3\sqrt{2}}{4\sqrt{3}-3\sqrt{2}}=a+b\sqrt{6}\\ & \Rightarrow \frac{\left(4\sqrt{3}+5\sqrt{2}\right)\left(4\sqrt{3}-3\sqrt{2}\right)}{48-18}=a+b\sqrt{6}\\ & \Rightarrow \frac{8\sqrt{6}+18}{30}=a+b\sqrt{6}\\ & \Rightarrow \frac{8\sqrt{6}}{30}+\frac{18}{30}=a+b\sqrt{6}\\ & \Rightarrow \frac{4}{15}\sqrt{6}+\frac{3}{5}=a+b\sqrt{6}\\ & \Rightarrow \frac{3}{5}+\frac{4}{15}\sqrt{6}=a+b\sqrt{6}\end{array}$$
By comparing coefficients of rational and irrational parts.
$$\begin{array}{rl}& \Rightarrow a=\frac{3}{5}\text{ , }b=\frac{4}{15}\\ & \therefore \left(\frac{3}{5},\frac{4}{15}\right)\end{array}$$