Two circles in the first quadrant of radii r 1 and r 2 touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line x + y = 2 . Then r 1 2 + r 2 2 - r 1 r 2 is equal to ____.

Step-by-step solution

Assuming a circle of radius r which touches both the axis is given by, x - r 2 + y - r 2 = r 2 Now given line x + y = 2 makes the intercept 2 the circle, so plotting the diagram we get, Now from diagram, A C = B C = 1 ,   O C = r 2 - 1 Now using the formula of perpendicular distance of point from the line we get, O C = 2 r - 2 2 = r 2 - 1 ⇒ 2 r - 1 2 = r 2 - 1 ⇒ r 2 - 4 r + 3 = 0 ⇒ r 1 = 1 ,   r 2 = 3 Hence, r 1 2 + r 2 2 - r 1 r 2 = 1 + 9 - 3 = 7