Question

Find a polar equation for the curve represented by the given Cartesian equation.

\({\rm{x y = 4}}\)

Short answer

The curve's polar equation is

\({{\rm{r}}^{\rm{2}}}{\rm{ = 8csc2\theta }}\)

Step-by-step solution

The curve's polar equation.

\(\begin{aligned}{l}{\rm{x = rcos\theta ,}}\;\;\;{\rm{y = rsin\theta }}\\{\rm{xy = 4(rcos\theta )(rsin\theta ) = 4}}\\{{\rm{r}}^{\rm{2}}}{\rm{cos\theta sin\theta = 4}}\\{{\rm{r}}^{\rm{2}}}{\rm{ = }}\frac{{\rm{4}}}{{{\rm{cos\theta sin\theta }}}}\\{{\rm{r}}^{\rm{2}}}{\rm{ = }}\frac{{\rm{8}}}{{{\rm{sin2\theta }}}}\end{aligned}\)

As a result, the curve's polar equation is.

It's important to understand that r is not the same as radius; it's simply a parameter that can take any actual value.

\({{\rm{r}}^{\rm{2}}}{\rm{ = 8csc2\theta }}\)