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        riddles >> medium >> Circles in a corner
        
(Message started by: Noke Lieu on Jan 21^st, 2008, 10:10pm)

Title: Circles in a corner
Post by Noke Lieu on Jan 21^st, 2008, 10:10pm

Whilst wrestling with somethingelse, this presented itself to me.

Take two circles of different radii and back them into a corner (as per the diagram)

What's the question though? That's something that's eluding me. I guess I'll settle for
The ratio of AB:BC

(because I haven't made it look particularly pretty, and hope someone around here can :-* )

Title: Re: Circles in a corner
Post by Obob on Jan 22^nd, 2008, 12:46am

[hide]It is not too difficult to find the coordinates of the points of intersection of the two circles, from which a formula for the ratio follows.[/hide]

Title: Re: Circles in a corner
Post by towr on Jan 22^nd, 2008, 1:12am

If the ratio is constant, it seems [hide]3+2sqrt(2)[/hide]

Title: Re: Circles in a corner
Post by Obob on Jan 22^nd, 2008, 10:05am

The way I interpreted the problem, the radi of the two circles are arbitrary. The circles are not necessarily orthogonal. And thus the ratio is not constant.

Title: Re: Circles in a corner
Post by towr on Jan 22^nd, 2008, 10:23am

Actually, I'm not assuming the ratio would be constant because supposedly the circles are orthogonal. If the ratio isn't constant regardless, I doubt the value I found is the ratio in the case of orthogonal circles (because that's not the case I used to calculate it).

Title: Re: Circles in a corner
Post by Obob on Jan 22^nd, 2008, 10:29am

Can we get some clarification as to where the point B is? Is it on the y-axis or is it the point where the two circles meet?

My calculations were based on B being on the y-axis.

Title: Re: Circles in a corner
Post by Noke Lieu on Jan 22^nd, 2008, 3:22pm

on 01/22/08 at 10:29:15, Obob wrote:

Can we get some clarification as to where the point B is? Is it on the y-axis or is it the point where the two circles meet?

My calculations were based on B being on the y-axis.

I intended B to be on the y axis.

But your question makes me think of the lengths of the arcs.

I am curious about how you guys went about determining where the circles intercept. I went about it in a seemingly obvious fashion, then I noticed something cuter.
(Doesn't quite work when there is no intercept though)

Title: Re: Circles in a corner
Post by Obob on Jan 22^nd, 2008, 4:04pm

I just took the equations (x-R)^2+(y-R)^2 = R^2 and (x-r)^2+(y-r)^2=r^2. Subtract the second equation from the first to get a linear equation relating x and y; solve for one of them and put it into the first equation to get a quadratic in x, say. Then use the quadratic formula to solve for x, and solving for y in the second equation gives the four solutions (x,y).

Title: Re: Circles in a corner
Post by Noke Lieu on Jan 22^nd, 2008, 7:48pm

Exactly what I did.
Then I noticed that the result of
(x-R)^2+(y-R)^2 = R^2 minus
(x-r)^2+(y-r)^2=r^2

could be re-arranged to y= -x + (R+r)/2

(so on the line that is perpendicular to the line joining centres; intercepting the x and y axes at the average of the two radii)

obvious when one thinks about it.