Three "touchy" circles

Geometry Level 4

Three circles with radii $r_1,r_2,r_3$ (where $r_1<r_2<r_3$ ) touch each other externally.If they have a common tangent,the value of $\sqrt{\frac{r_1}{r_2}} +\sqrt{\frac{r_1}{r_3}}$ is?

\sqrt2

2

1

2\sqrt2

1 solution

Sanchayan Dutta
Sep 15, 2015

The trick is to choose reference axes such that the common tangent coincides with the x-axis and the y-axis passes through the centre of the circle with radius $r_1$ !

$C_2=(a,r_2)$ $C_3=(b,r_3)$

Using $O_1O_2=r_1+r_2=\sqrt{a^2+(r_1-r_2)^2}$ we get $a=2\sqrt{r_1r_2}$ . Similarly get $b=-2\sqrt{r_1r_3}$ .

Use $O_2O_3=r_2+r_3$ to get $(a-b)^2=4r_2r_3$ and put the values of a and b as obtained above.Get the answer 1.Hurray!

$\color{#3D99F6}{\text{There is no space for solution. ,!!!} }$ Any way the solution.
$r_1 \color{#D61F06}{\text{ is the smallest. So each ratio is MUCH smaller than 1.} } \\ \color{#D61F06}{\text{ So only possible solution out of the given one is 1 . } }$

Niranjan Khanderia - 5 years, 8 months ago

Can you please explain how you get b <0 ? I know it has to be b<0, But it would be better if show these steps in detail. Thanks.

Niranjan Khanderia - 5 years, 8 months ago

I am getting answer 3+√3

SwayamS Mohapatra - 4 years, 3 months ago

Three "touchy" circles

1 solution

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