Q 36. - Calculus

Question

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

$r=2 \cos \theta$

Step-by-Step Solution

Answer

The required equation is ${\left(x-1\right)}^2+y^2=1$.

Step 1. Given information.

The given equation in polar coordinates is:

$r=2 \cos \theta$

Step 2. Find the equation in rectangular coordinates.

$r=2 \cos \theta$

First, multiply both sides by r,

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

Now add $-2x$ on both sides of the equation,

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

Complete the square in x,

${\left(x-1\right)}^2+y^2-1=0$

now add 1 on both sides of the equation,

$$\begin{gathered} {\left(x-1\right)}^2+y^2-1+1=0+1 \\ {\left(x-1\right)}^2+y^2=1 \end{gathered}$$

Therefore, the equation in rectangular coordinates is ${\left(x-1\right)}^2+y^2=1$.