Systems of Equations · Challenge problem

For every pair of numbers $x$ and $y$ with $x + y = u$ and $xy = v$, express each of the following in terms of $u$ and $v$ alone.

$(a)\ \dfrac{1}{x} + \dfrac{1}{y}$ $(b)\ \dfrac{x}{y} + \dfrac{y}{x}$ $(c)\ x^2 + y^2$