The points \(S\), \(T\), \(U\) and \(V\) have coordinates \((s,ms)\), \((t,mt)\), \((u,nu)\) and \((v,nv)\) respectively. The lines \(SV\) and \(UT\) meet the line \(y = 0\) at the points with coordinates \((p,0)\) and \((q,0)\), respectively. Show that \[p = \frac{(m - n)sv}{ms - nv},\] and write down a similar expression for \(q\).
Given that \(S\) and \(T\) lie on the circle \(x^2 + (y - c)^2 = r^2\), find a quadratic equation satisfied by \(s\) and by \(t\), and hence determine \(st\) and \(s + t\) in terms of \(m\), \(c\) and \(r\).
Given that \(S\), \(T\), \(U\) and \(V\) lie on the above circle, show that \(p + q = 0\).
