A circle whose radius is 5 and which touches externally the circle x2 + y2 – 2x – 4y – 20 = 0 at the point (5, 5) intersects in real distinct points the line

Centre of the given circle is A (1, 2) and its radius is 1+(2)2+20=5. Point of contact P is (5, 5). Let B (h, k) be the centre of the required circle of radius 5, then P is the mid-point of AB, so thath+12=5  and   k+22=5⇒h=9,k=8and an equation of the required circle is     x2+y2−18x−16y+120=0If x=0, y2−16y+120=0 does not give real values of yIf y=0,x2−18x+120=0 does not give real values of xIf          y=x,2x2−34x+120=0or   x2−17x+60=0  ⇒     x=5,12This shows that the circle intersects the line y = x at two real distinct points.