Question

The letters x and y represent rectangular coordinates. Write each equation using polar coordinates $\left(r,\theta \right)$.

$2xy=1$

Short answer

The equation is written using polar coordinates as $r^2\sin \left(2\theta \right)=1$.

Step-by-step solution

Step 1. Concept used

If P is a point with polar coordinates $(r,\theta )$, the rectangular coordinates $\left(x,y\right)$ of P are given by

$$\begin{gathered} x=r\cos \theta \\ y=r\sin \theta \end{gathered}$$

Step 2. Convert into polar coordinates

In the given equation $2xy=1$, we substitute $r\cos \theta$for $x$and $r\sin \theta$for $y$we get

role="math" localid="1646741958517" $\begin{array}{rcl}2xy& =& 1\\ 2\left(r\cos \theta \right)\left(r\sin \theta \right)& =& 1\\ 2r^2\cos \theta \sin \theta & =& 1\end{array}$

Now using the double angle formula $\sin \left(2\theta \right)=2\sin \theta \cos \theta$the equation can be written as

$\begin{array}{rcl}2r^2\cos \theta \sin \theta & =& 1\\ r^2\left(2\sin \theta \cos \theta \right)& =& 1\\ r^2\sin \left(2\theta \right)& =& 1\end{array}$