$\mathrm{Let} {\mathrm{\alpha }}_1 \mathrm{and} {\mathrm{\alpha }}_2$ be the ordinates of two points A and B on a parabola ${\mathrm{y}}^2=4\mathrm{ax} \mathrm{and} \mathrm{let} {\mathrm{\alpha }}_3$ be the ordinate of the point of intersection of its tangents at A and B . Then ${\mathrm{\alpha }}_3-{\mathrm{\alpha }}_2=$

$$\begin{gathered} \mathrm{Let} \mathrm{the} \mathrm{coordinates} \mathrm{of} \mathrm{P} \mathrm{and} \mathrm{Q} \mathrm{be} ({{\mathrm{at}}_1}^2,2{\mathrm{at}}_1) \mathrm{and} ({{\mathrm{at}}_2}^2,2{\mathrm{at}}_2) \mathrm{respectively}. \\ \mathrm{Then} {\alpha }_1=2{\mathrm{at}}_1\mathrm{and} {\alpha }_2=2{\mathrm{at}}_2. \\ \mathrm{The} \mathrm{coordinates} \mathrm{of} \mathrm{the} \mathrm{point} \mathrm{of} \mathrm{intersection} \mathrm{of} \mathrm{the} \mathrm{tangents} \mathrm{at} \mathrm{P} \mathrm{and} \mathrm{Q} \mathrm{are} \\ ({\mathrm{at}}_1{\mathrm{t}}_2,\mathrm{a}({\mathrm{t}}_1+{\mathrm{t}}_2\left)\right). \\ \therefore {\alpha }_3=\mathrm{a}({\mathrm{t}}_1+{\mathrm{t}}_2) \\ \mathrm{Now} {\mathrm{\alpha }}_3-{\mathrm{\alpha }}_2={\mathrm{at}}_1-{\mathrm{at}}_2={\mathrm{\alpha }}_1-{\mathrm{\alpha }}_3 \end{gathered}$$