if the axis of parabola y²=4x touches a circle of radius r such that the parabola cuts the circle at two distinct points at A and B and AB is the diameter of the circle then slope of AB is ?

general point on a parabola with eq y² =4ax  is  (at² ,2at)

but we have eq y² =4x

 so a =1

   so genral point is (t² ,2t) ..............1

parabola cuts the circle at two different points ,let these points be A,B..

A=(t₁²,2t₁)  & B=(t_2²,2t2)

circle touches the x axis so the y coordinate of its center is +r when circle is drawn above x axis & -r when drawn below..

since AB is diameter then

    coordinate of center of circle=A+B/2=[(t_{1² +t}2²)/2 ,(t_1+t2)]

  y coordinate of center =+ (r) or -(r)

            so t₁+t₂=+ (r) or -(r)         .................2

slope of AB=2(t₂-t_1)/(t_2²-t_1²)

                m=2/(t₁+t₂₎

               m=2/r ,-2/r