Chapter 1.1, Problem 22P

We call a coordinate system $\left(u,v\right)$ orthogonal if its coordinate curves (the two family of curves $\mathrm{u}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{v}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$) are orthogonal trajectories (for example, a Cartesian coordinate system or a polar co-ordinate system). Let $\left(u,v\right)$ be orthogonal coordinates, where $u=x^2+2y^2$, and x and y are Cartesian coordinates. Find the Cartesian equation of the v-coordinate curves, and sketch the $\left(u,v\right)$ coordinate system.