I'm having trouble with the second part of this question.
I can do the first part by finding the normal at P and where it intercepts with U and then for the second part i've substituted each point into the first equation...I don't know where to go from here, do i do simultaneous equations?
The equation in part (i) is satisfied by all three parameter values $p$, $q$ and $r$, so you need to write down the sum of the roots of this cubic equation.
A cubic of the form $ax^3+bx^2+cx+d=0$ whose roots are $\alpha,\beta, \gamma$ has sum of roots $\alpha+\beta+\gamma=-\frac ba$