The equation of the line passing through the points $\left({\mathrm{at}}_1^2,2{\mathrm{at}}_1\right),\left({\mathrm{at}}_2^2,2{\mathrm{at}}_2\right)\text{ is }$

slope of $\overrightarrow{\mathrm{AB}}\text{ is }\frac{2{\mathrm{at}}_2-2{\mathrm{at}}_1}{{\mathrm{at}}_2^3-{\mathrm{at}}_1^2}=\frac{2\mathrm{a}\left({\mathrm{t}}_2-{\mathrm{t}}_1\right)}{\mathrm{a}\left({\mathrm{t}}_2^2-{\mathrm{t}}_1^2\right)}=\frac{2\mathrm{a}\left({\mathrm{t}}_2-{\mathrm{t}}_1\right)}{\mathrm{a}\left({\mathrm{t}}_2-{\mathrm{t}}_1\right)\left({\mathrm{t}}_2+{\mathrm{t}}_1\right)}=\frac{2}{{\mathrm{t}}_2+{\mathrm{t}}_1}$

Equation to $\overrightarrow{\mathrm{AB}}\text{ is }\mathrm{y}-2{\mathrm{at}}_1=\frac{2}{{\mathrm{t}}_1+{\mathrm{t}}_2}\left(\mathrm{x}-{\mathrm{at}}_1^2\right)\Rightarrow \mathrm{y}\left({\mathrm{t}}_1+{\mathrm{t}}_2\right)-2{\mathrm{at}}_1\left({\mathrm{t}}_1+{\mathrm{t}}_2\right)=2\left(\mathrm{x}-{\mathrm{at}}_1^2\right)$

$\Rightarrow \mathrm{y}\left({\mathrm{t}}_1+{\mathrm{t}}_2\right)-2{\mathrm{at}}_1^2-2{\mathrm{at}}_1{\mathrm{t}}_2=2\mathrm{x}-2{\mathrm{at}}_1^2\Rightarrow \mathrm{y}\left({\mathrm{t}}_1+{\mathrm{t}}_2\right)=2\mathrm{x}+2{\mathrm{at}}_1{\mathrm{t}}_2$