Find natural no. x and y.

[jagdish]

Find all Natural no. $x$ and $y$ which are Co-prime and satisfy the relation $29.(x+y)+81xy=27(x^2+y^2)$

[nayel]

If $x=y$ then $x=y=1$, absurd. So $x\neq y$. We have
\[29(x+y)+81xy=27(x^2+y^2)> 54xy \Rightarrow 29(x+y)>27xy\Rightarrow \frac 1x+\frac 1y>\frac{27}{29}.\]
Hence $2\max\{1/x,1/y\}>27/29$ implying $\min\{x,y\}\le 2$. The rest is just casework.