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 \(q\).

Could we draw a sketch to keep track of what is going on?

Could we work out the equations of the straight lines involved in this problem?

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 \(r\).

What are the coordinates of \(S\)?

If we have a quadratic equation whose roots are \(s\) and \(t\), how are \(st\) and \(s + t\) related to the coefficients of this equation?