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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=(t22,2t2)
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=[(t12 +t22)/2 ,(t1+t2)]
y coordinate of center =+ (r) or -(r)
so t1+t2=+ (r) or -(r) .................2
slope of AB=2(t2-t1)/(t22-t12)
m=2/(t1+t2)
m=2/r ,-2/r
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