The tangents to a parabola at the vertex $$V$$ and any point $$P$$ meet at $$Q.$$ If $$S$$ be the focus then $$SP,\,SQ,\,SV$$    are in :

Solution :
Let the parabola be $${y^2} = 4ax.\,\,Q$$    is the intersection of the lines $$x = 0$$  and $$ty = x + a{t^2},$$    where $$P = \left( {a{t^2},\,2at} \right).$$    Solving these, $$Q = \left( {0,\,at} \right).$$   Also $$S = \left( {a,\,0} \right).$$
$$\eqalign{ & \therefore \,\,S{P^2} = {\left\{ {a\left( {{t^2} - 1} \right)} \right\}^2} + 4{a^2}{t^2} = {a^2}{\left( {{t^2} + 1} \right)^2} \cr & \,\,\,\,\,\,\,\,S{Q^2} = {a^2} + {a^2}{t^2} = {a^2}\left( {{t^2} + 1} \right){\text{ and }}SV = a \cr & \therefore \,S{Q^2} = SP.SV\, \cr} $$