Length of tangents from a point  $\left(x_1,y_1\right)$  to a circle  $x^2+y^2+2gx+2fy+c=0$  is given by
$\sqrt{x_1^2+y_1^2+2g{x}_1+2f{y}_1+c}$
The equation of the circle is, $x^2+y^2-4x-2y-11=0$
$\therefore$ its centre is $\left(2,1\right)$ ,
and radius $=\sqrt{4+1+11}=4=BC$

Length of the tangent from the point  $A(4, 5)$ is
$=\sqrt{16+25-16-10-11}=\sqrt{4}=2=AB$
$\therefore$ Area of quadrilateral $ABCD$ is

$=2($Area $\Delta ABC)=2\times \left(\frac{1}{2}AB\times BC\right)$

$=2\times \left(\frac{1}{2}\times 2\times 4\right)=8 \mathrm{square}$ units