Question

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

$4x^{2}y=1$

Short answer

The equation is written using polar coordinates as$4r^3{\cos }^2\theta \sin \theta =1$.

Step-by-step solution

Step 1. Concept used 

If P is a point with polar coordinates $\left(r,\theta \right)$, the rectangular coordinates $(x,y)$ 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 $4x^{2}y=1$, we substitute $r\cos \theta$for $x$and $r\sin \theta$for $y$we get

$\begin{array}{rcl}4x^{2}y& =& 1\\ 4{\left(r\cos \theta \right)}^{2}r\sin \theta & =& 1\\ 4r^2{\cos }^2\theta ·r\sin \theta & =& 1\\ 4r^3{\cos }^2\theta \sin \theta & =& 1\end{array}$