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Parametrizing a Circle - Problem 2
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Let’s try a harder example. Here, we are asked to parameterize the circle (x minus 3) squared, plus (y minus 4) squared equals 25 in the counter-clockwise direction. Now here is our circle and this is different from the problem we did before. Now the center is 3,4 and the radius is 5.

So let’s imagine how we would do this problem if the center were at the origin. So if the center were 0,0, I would use parametric equations x equals 5 cosine theta, and y equals 5 sine theta. Those would give me a circle traced out counter-clockwise, with radius 5. And I should also add in theta is between 0 and 2 pi, that will give me one complete circle. How do I get the circle that’s radius 5, centered at the origin, shifted up into the right, 3 inches to the right 4 units up. How do I do that?

It’s actually really easy to do that to parametric equations. Because they are written in terms of x and y, in order to shift to the right 3 units, all I have to do is add 3 to the x equation. So x equals 5 cosine theta plus 3. And to shift up 4 units all have to do is add 4 to the y equation. So y equals 5 sine theta plus 4.

I need the same domain restriction theta is between 0 and 2 pi. But this is how I get the parametric equations for the circle centered to (3,4) with radius 5.

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