Question 17
In the diagram below, the circles $\Gamma_1$ and $\Gamma_2$ have centres $O_1$ and $O_2$ respectively. The lines $AB,CD$ and $EF$ are tangent to both circles, where the points of tangency are $A,B,C,D,P,Q$ as shown in the diagram below, and $CD$ perpendicular to $EF$. The circle $\Gamma_1$ is tangent to the line $EF$ at the point $P$ and $\Gamma_2$ is tangent to the line $EF$ at the point $Q$. Given that $PQ=11$, $O_1O_2=61$ and $EF=x$, find the value of $x$.
