Let the set of all values of r , for which the circles (x+1)²+(y+4)²=r² and x²+y²-4 x-2 y-4=0 intersect at two distinct points be the interval ( , ) . Then is equal to

Step-by-step solution

Circle 1: center C₁ = (-1, -4) , radius r . Circle 2: (x-2)^2 + (y-1)^2 = 9 , center C₂ = (2, 1) , radius 3 . Distance d = 9+25 = 34 . For two intersection points: |r-3| r + 3 > 34 r > 34 -3 and |r-3| r ( 34 -3, 34 +3) , so = 34 -3 , = 34 +3 . = ( 34 )^2 - 9 = 34 - 9 = 25 .