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

Elissa89

Member
Oct 19, 2017
52
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

Administrator
Staff member
Feb 24, 2012
13,735
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).