# Convert r=7cos(theta) into a rectangular equation

#### Elissa89

##### Member
So we're learning to plot polar equations, which easy enough. But I got a question in the homework that wasn't covered in class:

Convert r=7cos(theta) into a rectangular equation. Use x and y values. I know how to convert when it's x=r*cos(theta) or y=r*sin(theta) and r and theta is given. But this is different and I don't know how to do it.

#### MarkFL

Staff member
Okay, we are given the polar equation:

$$\displaystyle r=6\cos(\theta)$$

Now, from:

$$\displaystyle x=r\cos(\theta)\implies \cos(\theta)=\frac{x}{r}$$

We may write:

$$\displaystyle r=6\left(\frac{x}{r}\right)$$

Multiply through by $$r$$:

$$\displaystyle r^2=6x$$

We know:

$$\displaystyle r^2=x^2+y^2$$

Hence, we have:

$$\displaystyle x^2+y^2=6x$$

This would technically suffice, but I would prefer to continue and put into standard form:

$$\displaystyle x^2-6x+y^2=0$$

Complete the square on $$x$$:

$$\displaystyle (x-6x+9)+y^2=9$$

$$\displaystyle (x-3)^2+y^2=3^2$$

Now it's easy to see we have a circle of radius 3 units centered at (3,0).