Tangents are drawn to a unit circle with centre at the origin from each point on the line $x+y=2.$ Then, the area (in sq. units) of the locus of the mid-point of the chord of contact is

$\left(x_1,y_1\right)$ lies on $x+y=2$
$\Rightarrow x_1+y_1=2\dots \left(\mathrm{i}\right)$
Chord of contact with respect to $\left(x_1,y_1\right)$
$x{x}_1+y{y}_1=1$

Also equation of chord whose mid point is $\left(h,k\right)$
$h^2+k^2=hx+ky$
$\therefore \frac{x_1}{h}=\frac{y_1}{k}=\frac{1}{h^2+k^2}$
$\Rightarrow x_1=\frac{h}{h^2+k^2}, y_1=\frac{k}{h^2+k^2}$
Substituting in $\left(\mathrm{i}\right)$, we get,
$\frac{h}{h^2+k^2}+\frac{k}{h^2+k^2}=2$
Hence, the required locus is $x^2+y^2=\frac{x}{2}+\frac{y}{2}$

The area of the circle is $\frac{\pi }{8}$ sq. units