Two circles \omega_{1} and \omega_{2} are said to be orthogonal if they intersect each other at right angles. In other words, for any point P lying on both \omega_{1} and \omega_{2}, if \ell_{1} is the line tangent to \omega_{1} at P and \ell_{2} is the line tangent to \omega_{2} at P, then \ell_{1} \perp \ell_{2}. (Two circles which do not intersect are not orthogonal.)
Let \triangle A B C be a triangle with area 20. Orthogonal circles \omega_{B} and \omega_{C} are drawn with \omega_{B} centered at B and \omega_{C} centered at C. Points T_{B} and T_{C} are placed on \omega_{B} and \omega_{C} respectively such that A T_{B} is tangent to \omega_{B} and A T_{C} is tangent to \omega_{C}. If A T_{B}=7 and A T_{C}=11, what is \tan \angle B A C?