A rectangle $R$ with end points of the one of its sides as $\left(1,2\right)$ and $\left(3,6\right)$ is inscribed in a circle. If the equation of a diameter of the circle is $2x-y+4=0$, then the area of $R$ is _____.

Equation of line $AB$ will be 

$y-2=2\left(x-1\right)$$\Rightarrow 2x-y=0$

Now the given diameter is parallel to $AB$

Then, the distance between diameter and line $AB$

$=\left(\frac{4}{\sqrt{2^2+1^2}}\right)=\frac{4}{\sqrt{5}}$

Thus, $BC=2\times \frac{4}{\sqrt{5}}=\frac{8}{\sqrt{5}}$

Also $AB=\sqrt{{\left(1-3\right)}^2+{\left(2-6\right)}^2}=\sqrt{20}=2\sqrt{5}$

Area of the rectangle$=AB\times BC=\frac{8}{\sqrt{5}}\times 2\sqrt{5}=16$