Please solve 1st question

Dear Student,

$$\begin{gathered} a+b=\frac{a}{b}+\frac{b}{a} \\ \Rightarrow a+b=\frac{a^2+b^2}{ab} \\ \Rightarrow a^2+b^2=ab\left(a+b\right) \\ \Rightarrow a^2+b^2=a^{2}b+a{b}^2 \\ \Rightarrow a^2-a^{2}b=a{b}^2-b^2 \\ \Rightarrow a^2\left(1-b\right)=b^2\left(a-1\right) \\ \Rightarrow a^2\left(b-1\right)+b^2\left(a-1\right)=0 ...\left(1\right) \\ Since a and b are positive integers \\ \Rightarrow a>0 and b>0 equivalently, a\ge 1 and b\ge 1 \\ \Rightarrow a^2>0 and b^2>0 and a-1\ge 0 and b-1\ge 0 \\ Thus for \left(1\right) to be true, \\ b-1=0 and a-1=0 \\ \Rightarrow a=1 and b=1 \\ Thus, a^2+b^2=1+1=2 \end{gathered}$$

Hope this information will clear your doubts about topic.

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