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March 18th, 2014, 08:05 AM   #1
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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.
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March 18th, 2014, 08:32 AM   #2
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Re: Circle tangent to parabola

Quote:
Originally Posted by Gluon
I have to show that:
Are you sure it's not

?

Quote:
Originally Posted by Gluon
and that Xp is a solution of:
Are you sure it's not

?

Do you know how to find a tangent vector at an then how to turn it 90 degrees counterclockwise?
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March 18th, 2014, 09:16 AM   #3
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Re: Circle tangent to parabola

Quote:
Originally Posted by Evgeny.Makarov
Quote:
Originally Posted by Gluon
I have to show that:
Are you sure it's not

?

Quote:
Originally Posted by Gluon
and that Xp is a solution of:
Are you sure it's not

?

Do you know how to find a tangent vector at an then how to turn it 90 degrees counterclockwise?
Yes, indeed you're right but I can't find the edit button to change my post. I know how to find a tangent line to a circle with implicit differentiation, but I don't know how to proceed. Any suggestions?
Thanks from Rishabh11199
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March 18th, 2014, 09:30 AM   #4
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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 .
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March 18th, 2014, 10:28 AM   #5
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Re: Circle tangent to parabola

Wow great, thanks!
1. Tangent vector at : .
2. Perpendicular unit vector:
3. Thus:
Should be good, right?
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March 18th, 2014, 10:38 AM   #6
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Re: Circle tangent to parabola

Quote:
Originally Posted by Gluon
1. Tangent vector at : .
and are parallel vectors, but they are not equal.
Quote:
Originally Posted by Gluon
3. Thus:
In the second equation, x should be replaced by . Otherwise, this is correct.
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March 18th, 2014, 11:21 AM   #7
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Re: Circle tangent to parabola

Ok I get it. Thanks for the effort and help
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