Q 37. - Calculus

Question

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates. 

$r=5 \sin \theta$

Step-by-Step Solution

Answer

The required equation is $x^2+{\left(y-\frac{5}{2}\right)}^2={\left(\frac{5}{2}\right)}^2$.

Step 1. Given information.

The given equation in polar coordinates is:

$r=5 \sin \theta$

Step 2. Find the equation in rectangular coordinates.

$r=5 \sin \theta$

First, multiply both sides by r,

$$\begin{gathered} r^2=5r \sin \theta \\ {r}^2=5y \left[\mathrm{Since} r \sin \theta =y\right] \\ {x}^2+y^2=5y \left[\mathrm{Since} {\mathrm{r}}^2={\mathrm{x}}^2+{\mathrm{y}}^2\right] \end{gathered}$$

Now add $-5y$ on both sides of the equation,

$$\begin{gathered} x^2+y^2-5y=5y-5y \\ {x}^2+y^2-5y=0 \end{gathered}$$

Complete the square in y,

$x^2+{\left(y-\frac{5}{2}\right)}^2-{\left(\frac{5}{2}\right)}^2=0$

now add ${\left(\frac{5}{2}\right)}^2$ on both sides of the equation,

$$\begin{gathered} x^2+{\left(y-\frac{5}{2}\right)}^2-{\left(\frac{5}{2}\right)}^2+{\left(\frac{5}{2}\right)}^2=0+{\left(\frac{5}{2}\right)}^2 \\ {x}^2+{\left(y-\frac{5}{2}\right)}^2={\left(\frac{5}{2}\right)}^2 \end{gathered}$$

Therefore, the equation in rectangular coordinates is $x^2+{\left(y-\frac{5}{2}\right)}^2={\left(\frac{5}{2}\right)}^2$.