Externally Tangent Circle

Question

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?

Answer 1

Hint:

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

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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.