# Find the length of AB

In the diagram 4 circles of equal radius stand in a row in such a way that each circle touches the next one. $P$ is a point on the circumference of the first circle. The center of the fourth circle is point $Q$. The line $PQ$ goes through the centers of all four circles. $PC$ is a tangent on the fourth circle such that it intersects the second circle at points $A$ and $B$. Radius of the circles is $7$ and the length of $AB$ is $a\sqrt b$.
Find the length.

Let $\angle CPQ=\alpha$. We have : $\sin \alpha=\dfrac{CQ}{QP}=\dfrac{7}{7^2}= \dfrac{1}{7}$.
The distance from PC tho the center of the second circle is $d=3\cdot 7 \cdot \sin \alpha=3$ so the chord $AB$ is $AB=2\sqrt{7^2-3^2}=4\sqrt{10}$