Question

If $x,y>0,lo{g}_{y}x+lo{g}_{x}y=\frac{10}{3}$ and $xy=144$ , then $\frac{x+y}{2}=\sqrt{N}$, when N is a natural number , find the value of N.

Answer 1

$$\begin{array}{c}{\log }_{x}y=t\\ {\log }_{y}x+{\log }_{x}y=\frac{10}{3}\\ t+\frac{1}{t}=\frac{10}{3}\\ 3(t^2+1)-10t=0\\ 3t^{2}9t-t+3=0\\ 3t(t-3)-(t-3)=0\\ t=3ort=\frac{1}{3}\\ {\log }_{x}y=3;{\log }_{x}y=\frac{1}{3}\\ y=x^3;y=x^{\frac{1}{3}}\\ xy=144;xy=144\\ puty=x^{3}putx=y^3\\ x^{{}^4}=144y=2\sqrt{3}\\ x^2=12x=24\sqrt{3}\\ x=2\sqrt{3}\\ y=24\sqrt{3}\\ \frac{x+y}{2}=13\sqrt{3}\\ =\sqrt{507}\\ N=507\\ \end{array}$$