Let S be the set of all-natural number, for which the line $\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=2$ is tangent to the curve ${\left(\frac{{\displaystyle \mathrm{x}}}{{\displaystyle \mathrm{a}}}\right)}^{\mathrm{n}}+{\left(\frac{{\displaystyle \mathrm{y}}}{{\displaystyle \mathrm{b}}}\right)}^{\mathrm{n}}=2$ at the point (a, b), ab≠0 then

${\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^{\mathrm{n}}+{\left(\frac{\mathrm{y}}{\mathrm{b}}\right)}^{\mathrm{n}}=2$
Slope of the tangent (a, b)
$\frac{\mathrm{dy}}{\mathrm{dx}}]=\frac{-\mathrm{b}}{\mathrm{a}}$
$\begin{array}{l}\mathrm{Equation}\mathrm{of}\mathrm{tangent}\mathrm{y}-\mathrm{b}=\frac{-\mathrm{b}}{\mathrm{a}}(\mathrm{x}-\mathrm{a})\\ \frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=2,\forall \mathrm{n}\in \mathrm{N}\\ \mathrm{S}=\mathrm{N}\end{array}$