If $\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ and $\left({\mathrm{x}}_2,{\mathrm{y}}_2\right)$ are ends of focal chord of ${\mathrm{y}}^2=4\mathrm{ax}$ then ${\mathrm{x}}_1{\mathrm{x}}_2+{\mathrm{y}}_1{\mathrm{y}}_2=$
 

$\left({\mathrm{x}}_1, {\mathrm{y}}_1\right) \left({\mathrm{x}}_2, {\mathrm{y}}_2\right)$ ends of focal chord of

$$\begin{gathered} {\mathrm{y}}^2=4\mathrm{ax} \\ \left({\mathrm{x}}_1, {\mathrm{y}}_1\right)=\left({{\mathrm{at}}_1}^2, 2{\mathrm{at}}_1\right) \\ \left({\mathrm{x}}_2, {\mathrm{y}}_2\right)=\left({{\mathrm{at}}_2}^2, 2{\mathrm{at}}_2\right) \\ Since it is a focal chord then {\mathrm{t}}_1{\mathrm{t}}_2=-1 \\ {\mathrm{x}}_1 {\mathrm{x}}_2={\mathrm{a}}^2, {\mathrm{y}}_1 {\mathrm{y}}_2=-4{\mathrm{a}}^2 \\ Now {\mathrm{x}}_1 {\mathrm{x}}_2+{\mathrm{y}}_1 {\mathrm{y}}_2={\mathrm{a}}^2-4{\mathrm{a}}^2 \\ =-3{{\mathrm{a}}^2}^{ } \end{gathered}$$