Two circles with two common intersect points

Question

I have two circles which intersect and have two common intersections. The larger circle has radius 5 and the smaller has radius 3. The distance between the centre of the two centres of the circle is equal to 6. I don’t know the length of either tangents, how do I find the length from the centre of the larger circle (A), to the point where the two outer tangents intersect (B)?

Answer 1

Hint: Observe that point $B$ lies on the line connecting the centers of the circles. From the similar right triangles one obtains: $$ \frac{r_2-r_1}{d}=\frac{r_2}x\implies x=\frac{r_2d}{r_2-r_1}=\frac{5\cdot 6}{5-3}=15. $$ Here $r_2$ and $r_1$ are the radii of the small and large circles, respectively, $d$ is the distance between the centers of the circles, and $x$ is the distance in question.