Two circles `C_1a n dC_2` both pass through the points `A(1,2)a n dE(2,1)` and touch the line `4x-2y=9` at `B and D ,` respectively. The possible coordinates of a point `C ,` such that the quadrilateral `A B C D` is a parallelogram, are `(a ,b)dot` Then the value of `|a b|` is_________

Answer

4

Explanation

The radical axis bisects the common tangents BD.
Hence, M is the midpoint of BD.
Let `C -= (a,b)`

Now, `C(a,b)` lies on common chord AE which is `y-2 = -1(x-1)` or `x+y=3`. Therefore,
`a+b=3` (1)
Also, `M((a+1)/(2),(b+2)/(2))` lies on `4x-2y=9`. Therefore,
`4((a+1)/(2))-2 ((b+2)/(2))=9`
or `2a+2-b-2=9`
or `2a-b=9` (2)
Solving (1) and (2) , `a=4` and `b = -1`. Therefore,
`:. | ab| =4`