Chapter 10.5, Problem 2ES

Let an $x\text{'}y\text{'}\text{-coordinate}$ system be obtained by rotating an $xy\text{-coordinate}$ system through an angle of $\theta =30°.$

(a) Find the $x\text{'}y\text{'}\text{-coordinates}$ of the point whose $xy\text{-coordinates}$ are $\left(1,-\sqrt{3}\right).$

(b) Find an equation of the curve $2x^2+2\sqrt{3}xy=3\text{in}x\text{'}y\text{'}\text{-coordinates}\text{.}$

(c) Sketch the curve in part (b), showing both $xy\text{-axes}$ and $x\text{'}y\text{'}\text{-axes}\text{.}$