# Externally Tangent Circle [closed]

Two cirles, each of radius 3cm touch each other along a common tangent. In how many ways can a circle of radius 8cm touch both of the circle externally?

## closed as off-topic by Leucippus, Jean-Claude Arbaut, DonAntonio, Watson, ShaileshOct 15 '16 at 0:33

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• Do you have any thoughts about this yourself? Have you drawn a diagram? – Henning Makholm Oct 14 '16 at 18:55
• voting to close, this is a really low quality question – Alex Robinson Oct 14 '16 at 19:01

Hint:

use the figure and find the two centers $D$ and $H$ using the rectangular triangles $DCB$ and $HCB$.

The two circles of radius $3$ have centers $A=(3,0)$ and $B=(-3,0)$. If a circle is tangent to these two circles and its center is external to the two circles, than this center have to stay on the axis of $AB$. The figure illustrate this situation for two such circle. The blue, that contains the centers $A$ and $B$ and has center $D=(0,d)$, and the red that does not contains $A$ and $B$ and has center $H=(0,h)$.

From the figure we can see that: $$\overline{HB}^2=\overline{BC}^2+\overline{CH}^2$$ and, since $\overline{HB}=\overline{BK}+\overline{KH}=3+8$, we have:

$$11^2=3^2+h^2$$

solving or $h$ we have two solutions: one is the point $H$ with $h>0$ and the other is for the symmetric solution with respect to the $x$ axis.

In the same way we can find the point $D$ and its symmetric.

• Well. Obviously the externally tangent is the red circle and there is a symmetric solution with respect to the $x$ axis. I simply give a hint suggesting a geometric-analytic approach (perhaps too brief). – Emilio Novati Oct 15 '16 at 9:51
• I am a class 9 boy and so am not sure what is meant by those hints – SayDude Oct 16 '16 at 10:01
• @SayDude: I've added something to my answer. I hope it's useful. – Emilio Novati Oct 16 '16 at 15:08