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

$$\begin{gathered} Given parabola y^2=4ax \\ Let A\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)=\left({\mathrm{at}}^2, 2\mathrm{at}\right) \\ and B\left({\mathrm{x}}_2,{\mathrm{y}}_2\right)=\left(\frac{\mathrm{a}}{{\mathrm{t}}^2},\frac{-2\mathrm{a}}{\mathrm{t}}\right) are end points of focal chord \\ Now {\mathrm{x}}_1{\mathrm{x}}_2+{\mathrm{y}}_1{\mathrm{y}}_2=\left({\mathrm{at}}^2\right)\left(\frac{\mathrm{a}}{{\mathrm{t}}^2}\right)+\left(2\mathrm{at}\right)\left(\frac{-2\mathrm{a}}{\mathrm{t}}\right) \\ ={\mathrm{a}}^2-4{\mathrm{a}}^2 \\ =-3{\mathrm{a}}^2 \end{gathered}$$