Let the set of all values of r, for which the circles (x+1)2+(y+4)2=r2 and x2+y2−4x−2y−4=0 intersect at two distinct points, be the interval (α,β). Then αβ is equal to:
Solutions · 1
Official solution
Second circle: centre (2,1), radius 3. Distance between centres:
\[d=\sqrt{(2-(-1))^2+(1-(-4))^2}=\sqrt{9+25}=\sqrt{34}.\]
For two distinct intersection points: ∣r−3∣<34<r+3, giving
\[\alpha=\sqrt{34}-3,\qquad \beta=\sqrt{34}+3,\qquad \alpha\beta = 34-9=25.\]