Two circles C, and C₂ both pass through the points A(l ,2) and E(2, 1) and touch the line  $4\mathrm{x}-2\mathrm{y}=9$ at B and D, respectively. The possible coordinates of a point C such that the quadrilateral ABCD is a parallelogram , are (a, b). Then the value of ab is _________

The radical axis bisects the common tangent BD.
Hence, M is the mid Point of BD.
Let $\mathrm{C}\equiv (\mathrm{a},\mathrm{b})$

C(a, b) lies on common chord AE which is $y-2=-1(x-1)$
or $\mathrm{x}+\mathrm{y}=3$ Therefore,
a + b = 3………………(i)
Also $\mathrm{M}\left(\frac{\mathrm{a}+1}{2},\frac{\mathrm{b}+2}{2}\right)$ lies on $4\mathrm{x}-2\mathrm{y}=9$ therefore,
$4\left(\frac{\mathrm{a}+1}{2}\right)-2\left(\frac{\mathrm{b}+2}{2}\right)=9$
or $2\mathrm{a}+2-\mathrm{b}-2=9$
or $2\mathrm{a}-\mathrm{b}=9$……….(ii)
Solving (i) and (ii), a = 4 and b = -1. Therefore
   ab= -4