March 18th, 2014, 08:05 AM  #1 
Newbie Joined: Mar 2014 Posts: 4 Thanks: 1  Circle tangent to parabola
Hi, I am stuck with some homework and I hope someone can help me. The problem is depicted in the attachment. A circle moves along a parabola. The purpose is to find the orbit of the center of the circle. The contact point of a circle and a parabola is and the center of the circle is . r and F are respectively the radius of the circle and the focus of the parabola. I have to show that: and that Xp is a solution of: I am stuck and would really appreciate if someone can help me. Thanks in advance. 
March 18th, 2014, 08:32 AM  #2  
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108  Re: Circle tangent to parabola Quote:
? Quote:
? Do you know how to find a tangent vector at an then how to turn it 90 degrees counterclockwise?  
March 18th, 2014, 09:16 AM  #3  
Newbie Joined: Mar 2014 Posts: 4 Thanks: 1  Re: Circle tangent to parabola Quote:
 
March 18th, 2014, 09:30 AM  #4 
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108  Re: Circle tangent to parabola
A tangent vector to the graph y = f(x) at point has slope . Thus, as a tangent vector itself you can take or any vector proportional to it. Next, if (a, b) is a vector, then the result of turning it by 90 degrees is (b, a). Indeed, their scalar product is a(b) + ba = 0, which means they are perpendicular. Finally, if is obtained by shifting in the direction of some vector (a, b) by distance r, then , and similarly for y. Indeed, is the unit vector pointing from to . So, take and find some tangent vector at (preferably, its coordinates should not have fractions for easier calculations). Then find a perpendicular vector. Find a unit perpendicular vector. Multiply it by r and use the result to shift to . 
March 18th, 2014, 10:28 AM  #5 
Newbie Joined: Mar 2014 Posts: 4 Thanks: 1  Re: Circle tangent to parabola
Wow great, thanks! 1. Tangent vector at : . 2. Perpendicular unit vector: 3. Thus: Should be good, right? 
March 18th, 2014, 10:38 AM  #6  
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108  Re: Circle tangent to parabola Quote:
Quote:
 
March 18th, 2014, 11:21 AM  #7 
Newbie Joined: Mar 2014 Posts: 4 Thanks: 1  Re: Circle tangent to parabola
Ok I get it. Thanks for the effort and help 

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