Find the sum of all real numbers r such that there is at least one point where the circle with radius r centered at (4,39) is tangent to the parabola with equation 2y=x2−8x+12.
Case 2: Since O lies on the axis of symmetry of the parabola yA=yB So if we write an equation in terms of y, the discriminant should be zero.
(x−4)2+(y−39)2=r22y=x2−8x+12⇒x2−8x+16+y2−78y+392=r2⇒2y+4+y2−78y+392=r2⇒y2−76y+392+4−r2=0D=Δ=0=(76)2−4(392+4−r2)÷4⇒0=382−392−4+r2r2=392−382+4=(39−38)(39+38)+4=77+4=81⇒r=9 By Hints 2 Ans=41+9=50
