Two circles in the first quadrant of radii r _ 1 and r _ 2 touch...

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,

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

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

$⇒ {r}^{2} - 4r + 3 = 0$

$⇒ {r}_{1} = 1, {r}_{2} = 3$

Hence, ${{r}_{1}}^{2} + {{r}_{2}}^{2} - {r}_{1}{r}_{2} = 1 + 9 - 3 = 7$