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, Shailesh Oct 15 '16 at 0:33

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

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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.

  • $\begingroup$ 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). $\endgroup$ – Emilio Novati Oct 15 '16 at 9:51
  • $\begingroup$ I am a class 9 boy and so am not sure what is meant by those hints $\endgroup$ – SayDude Oct 16 '16 at 10:01
  • $\begingroup$ @SayDude: I've added something to my answer. I hope it's useful. $\endgroup$ – Emilio Novati Oct 16 '16 at 15:08

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