2007 AJC P1 Q12

A curve is defined by the parametric equations

\(x=a{{t}^{2}},\,y=2at.\)

Show that the equation of the normal to the curve at the point \((a{{t}^{2}},\,2at)\) is at \(y+tx=a{{t}^{3}}+2at.\)

If the normal to the curve at the point \((a{{t}^{2}},\,2at)\) meets the curve again at the point \((a{{q}^{2}},\,2aq),\) show that \({{t}^{2}}+qt+2=0.\)

Deduce that \({{q}^{2}}\) cannot be less than \(8\).