Chapter H.1, Problem 13E

To determine

To identify: The curve by finding theCartesian equation for the curve

Answer to Problem 13E

  $x^2+{\left(y-\frac{3}{2}\right)}^2=\frac{9}{4}$ is the Cartesian equation and the curve is a circle with center $\left(0,\frac{3}{2}\right)$ , and radius $\frac{3}{2}$.

Explanation of Solution

Given:

  $r=3\sin \theta$

Formula Used:

  $x=r\cos \theta$ , $y=r\sin \theta$

Calculation:

We have $x=r\cos \theta$

  $y=r\sin \theta$

  $\Rightarrow \cos \theta =\frac{x}{r}$

  $\sin \theta =\frac{y}{r}$

  $\therefore r=3\sin \theta$ becomes

  $r=\frac{3y}{r}$

  $\Rightarrow r^2=3y$

  $\Rightarrow x^2+y^2=3y$

  $\Rightarrow x^2+y^2-3y=0$

  $\Rightarrow x^2+y^2-3y+{\left(\frac{3}{2}\right)}^2={\left(\frac{3}{2}\right)}^2$ [ $\because$ Both sides adding $\frac{9}{4}$ ]

  $\Rightarrow x^2+{\left(y-\frac{3}{2}\right)}^2=\frac{9}{4}$

  $\because$ The curve is a circle with center $\left(0,\frac{3}{2}\right)$ , and radius $\frac{3}{2}$

Conclusion:

Therefore, $x^2+{\left(y-\frac{3}{2}\right)}^2=\frac{9}{4}$ is the Cartesian equation and the curve is a circle with center $\left(0,\frac{3}{2}\right)$ , and radius $\frac{3}{2}$.