Problem

The equation of a curve is $y=\dfrac{1}{2}k^2x^2-2kx+2$ and the equation of a line is $y=kx+p$, where $k$ and $p$ are constants with $0<k<1$.

(a) It is given that one of the points of intersection of the curve and the line has coordinates $\left(\dfrac{5}{2},\dfrac{1}{2}\right)$.

Find the values of $k$ and $p$, and find the coordinates of the other point of intersection.

(b) It is given instead that the line and the curve do not intersect.

Find the set of possible values of $p$.