Two circles in the first quadrant of radii r 1 and r 2 touch the coordinate axes. Each of them cuts off an…

Assuming a circle of radius $r$ which touches both the axis is given by, ${\left(x-r\right)}^2+{\left(y-r\right)}^2=r^2$

Now given line $x+y=2$ makes the intercept $2$ the circle, so plotting the diagram we get,

Now from diagram,

$AC=BC=1, OC=\sqrt{r^2-1}$

Now using the formula of perpendicular distance of point from the line we get,

$OC=\left|\frac{2r-2}{\sqrt{2}}\right|=\sqrt{r^2-1}$

$\Rightarrow 2{\left(r-1\right)}^2=r^2-1$

$\Rightarrow r^2-4r+3=0$

$\Rightarrow r_1=1, r_2=3$

Hence,  ${r_1}^2+{r_2}^2-r_1{r}_2=1+9-3=7$