MHB Find $\frac{a}{b}$ in the Circle of Balls

[maxkor]

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How find $\frac{a}{b}$

 

[I like Serena]

How find $\frac{a}{b}$

Hi maxkor! (Smile)

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

 

[maxkor]

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Let 1/2b radius of the big circle, let r radius of the smaller circle
Let $c=1/2a=1/2b−r,
b=2c+2r,
a=2c.$
So $\frac{a}{b}=\frac{2c}{2c+2r}=\frac{c}{c+r}$
Small circles respectively tangential to the large circles so
$z=c+2r,t=a−r=2c−r$

Is this right?

 

[Opalg]

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Use Pythagoras in the triangles $CXY$, $DXY$ (where $Y$ is the centre of one of the footballs) to find two expressions for $XY^2$ in terms of $a$, $b$ and $r$. Putting those expressions equal to each other will give you an equation connecting $a$, $b$ and $r$.

You already know that $r = \frac12(b-a)$ (from your equation $c = \frac12a = \frac12b-r$). Substitute that value of $r$ into your equation, and it will give you the connection between $a$ and $b$.

 

[maxkor]

Is $\frac{a}{b}=\frac{\sqrt{2}}{2}$ correct answer?

 

[Opalg]

Is $\frac{a}{b}=\frac{\sqrt{2}}{2}$ correct answer?

Yes! (Happy)