Find the coordinates of the point at which the circles $${x^2}\, + \,{y^2} - \,4x - \,2y = - 4\,\,and\,\,{x^2}\, + \,{y^2} - \,12x - \,8y = - 36$$ touch each other. Also find equations common tangests touching the circles in the distinct points.

If $$A > 0,B > 0\,$$ and $$A + B = \pi /3,$$ then the maximum value of tan A tan B is _______.

Number of solutions of the equation $$\tan x + \sec x = 2\cos x\,$$ lying in the interval $$\left[ {0,2\pi } \right]$$ is:

$$ABCD$$ is a rhombus. Its diagonals $$AC$$ and $$BD$$ intersect at the point $$M$$ and satisfy $$BD$$ = 2$$AC$$. If the points $$D$$ and $$M$$ represent the complex numbers $$1 + i$$ and $$2 - i$$ respectively, then A represents the comp[lex number ..........or..........