Question

Two circles $C_1$ and $C_2$ both passes through the points $A(1,2)$ and $E(2,1)$ and touch the line $4x-2y=9$ at $B$ and $D$ respectively. The possible cordinates of a point $C$ such that the quadrilateral $ABCD$ is a parallelogram is $(a,b)$ then the value of $|ab|$ is

Answer 1

The radical axis bisects the common tangent $BD$. Hence, $M$ is midpoint of $BD$.
Let $C\equiv (a,b)$
Now, $C(a,b)$ lies on the common chord $AE$ which is $y-2=-1(x-1)$ or $x+y=3$.
Therefore, $a+b=3\left(i\right)$
Also, $M(\frac{a+1}{2},\frac{b+2}{2})$ lies on $4x-2y=9$. Therefore,
$4\left(\frac{a+1}{2}\right)-2\left(\frac{b+2}{2}\right)=9$
or $2a+2-b-2=9$
or $2a-b=9\left(ii\right)$
Solving, $\left(i\right)$ and $\left(ii\right)$, $a=4$ and $b=-1$, therefore,
$a+b=3$