Families of Polar Curves: Circles, Cardiods, and Limacon - Concept

I want to talk about a family of polar curves that's described by these 2 equations r=a+b cosine theta or r=a+b sine theta. Now when b equals 0, all you get is r=a, and this is going to be the graph of the circle. And it's easy to show that. Remember that r=a, you could square both sides and get r squared equals a squared. And r squared when you're converting to rectangular is x squared plus y squared. So this is a circle centered at the origin. And its radius is a.
Now what about the other simple case, when a equals 0. Well, if we get r=b cosine theta or r=b sine theta, I'll investigate this one. Its graph happens to also be a circle and that's also pretty easy to show. If you take r=b sine theta and multiply both sides by r you get r squared equals b times r sine theta.
Now converting back to rectangular r squared again becomes x squared plus y squared and this become b times r sine theta which is y. So b times y. And I'll collect all terms on the left. x squared plus y squared minus by. If you want to find out what the circle is, you can complete the square on y and remember to do that, you take half of this number -b over 2 and square it. So you add b squared over 4 to both sides. And then you get x squared plus and this factors has y-b over 2 quantity squared equals b squared over 4. And so you can see this also a circle centered at 0 b over 2 and its radius is also b over 2.
So let's look at examples of each of these 2 special cases of this family of polar curves. We've got a case where b=0, so r=6. I just wrote this in so you could see the family but this is basically r=6 in blue. A circle centered at the origin with radius 6. And r=0+8 sine theta. Here a=0 and b is 8. Notice that the radius is actually 4 and the center is right here at rectangular 0 4 in polar 4 pi over 2.
So just remember that the special cases of this family when either of these coefficients are 0, are circles. We'll investigate the other cases in a moment.