Chapter 9.1, Problem 72AYU

To determine

To find: Equation using polar coordinates $\left(r,\theta \right)$.The Letters $x$ and $y$ represent rectangular coordinates.

Answer to Problem 72AYU

Solution:

$2r^3\left(\text{sin}2\theta \right)=1$

Explanation of Solution

Given:

$4x^{2}y=1$

Calculation:

Use the formula,

$x=r\text{cos}\theta$ $y=r\text{sin}\theta$

We have,

$4x^{2}y=1$

$4{\left(r\text{cos}\theta \right)}^2\left(r\text{sin}\theta \right)=1$

$4r^2\text{cos}\theta r\text{sin}\theta =1$

Using $2\text{cos}\theta \text{sin}\theta =\text{ sin}2\theta$, we have

$2r^3\left(2\text{cos}\theta \text{sin}\theta \right)=1$

$2r^3\left(\text{sin}2\theta \right)=1$