Figure It Out.
Find the sum of all k such that such a circle C 3 could exist in the following arrangement:
Two circles C 1 , C 2 of radii 3 1 and 7 2 respectively touch each other externally and they both touch a unit circle C internally. A circle C 3 of radius R = k 2 is inscribed to touch the circles C 1 , C 2 externally and the circle C internally.
The answer is 22.
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1 solution
@Aaghaz Mahajan , whats that?
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Here..........see this link...........
https://brilliant.org/wiki/descartes-theorem/
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Yes, that is the theorem to use, but it yields a value of R = 3 2 and not 1 9 2 .
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@Brian Charlesworth – Umm, no sir........you might have taken the other value...........2/3 is the other value we get after solving the quadratic equation.........and to see which one is the correct value, we need to take into account the sign of the respective curvatures of the circles..........
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@Aaghaz Mahajan – Hmmm... Yes, the sign is critical, and with the exterior unit circle ascribed a curvature of k 1 = − 1 then the Descartes' formula is satisfied using values k 2 = 3 , k 3 = 3 . 5 and k 4 = 1 . 5 .
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@Brian Charlesworth – Yes sir, exactly!!! I think that was the mistake you might have committed earlier......
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@Aaghaz Mahajan – O.k., but the three circles C 1 , C 2 , C 3 inside C will then have respective radii of 3 1 , 7 2 and 3 2 . I don't know where the value of 1 9 2 comes from; with the radii I have mentioned the other circle touching each of C 1 , C 2 , C 3 has a radius of 1 7 1 , (the second solution to the Descartes' formula).
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@Brian Charlesworth – Ummm...........Sir, I don't se where you are going wrong.........this is a simple quadratic equation........Here.......The values of the respective curvatures are 3 , 3.5 and -1........the curvature which we need to find can be taken as x........Now, after applying Descarte's formula, we solve the equation to get the value of x as 9.5 and 1.5.........Now, we see that the circle with curvature 9.5 satisfies the constraint........and hence the radius is 1/(9.5) = 2/19............
@Brian Charlesworth – Well in response to my previous comment, I notice the flaw in the question.............there are actually TWO values of the radii, and there exist TWO unique circles as a solution the problem (drawing a diagram helps in visualising it)........ I am sorry for any trouble caused.......
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@Aaghaz Mahajan – Oh right, I see that now! So Priyanshu will have to specify which circle he means by C 3 , since at the moment both 3 2 and 1 9 2 are possible radii.
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@Brian Charlesworth – Yeah........I have filed a report........
A very simple application of Kissing Circle's theorem........!!!