Let C₁ be the circle in the third quadrant of radius 3 , that touches both coordinate axes. Let C ₂ be the circle with centre (1,3) that touches C ₁ externally at the point ( , ) . If ( - )^2= m n , gcd (m, n)=1 , then m+n is equal to :

Step-by-step solution

C₁:(x+3)^2+(y+3)^2=3^2 Let C ₁ and C ₂ has centres A (-3₁-3 ) and B(1,3) aligned & AB = 16+36 =2 13 & r ₁=3 and r ₂=2 13 -3 & P ( , ), = r ₁(1)+ r ₂(-3) r ₁+ r ₂ , = r ₁(3)+ r ₂(-3) r ₁+ r ₂ & = 3-3(2 13 -3) 2 13 , = 18-6 13 2 13 & ( - )^2= ( 6 2 13 )^2 & ( - )^2= ( 6 2 13 )^2, ~m + n =22 aligned