Find the equation of the straight line passing through the points $\left({\mathrm{at}}_1^2,2{\mathrm{at}}_1\right)\text{ and }\left({\mathrm{at}}_2^2,2{\mathrm{at}}_2\right)$.

The equation of the st.line passing through  the points $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)\text{ is }$
$\mathrm{y}-{\mathrm{y}}_1=\frac{{\mathrm{y}}_2-{\mathrm{y}}_1}{{\mathrm{x}}_2-{\mathrm{x}}_1}\left(\mathrm{x}-{\mathrm{x}}_1\right)$
Given points are $\left({\mathrm{at}}_1^2,2{\mathrm{at}}_1\right)\text{ and }\left({\mathrm{at}}_2^2,2{\mathrm{at}}_2\right)$
the eq’n of st. line is
$\begin{array}{l}\mathrm{y}-2{\mathrm{at}}_1=\frac{2{\mathrm{at}}_2-2{\mathrm{at}}_1}{{\mathrm{at}}_2^2-{\mathrm{at}}_1^2}\left(\mathrm{x}-{\mathrm{at}}_1^2\right)\\ \Rightarrow \mathrm{y}-2{\mathrm{at}}_1=\frac{2\mathrm{a}\left({\mathrm{t}}_2-{\mathrm{t}}_1\right)}{\mathrm{a}\left({\mathrm{t}}_2+{\mathrm{t}}_1\right)\left({\mathrm{t}}_2-{\mathrm{t}}_1\right)}\left(\mathrm{x}-{\mathrm{at}}_1^2\right)\\ \Rightarrow \mathrm{y}-2{\mathrm{at}}_1=\frac{2}{{\mathrm{t}}_2+{\mathrm{t}}_1}\left(\mathrm{x}-{{\mathrm{at}}_1}^2\right)\\ \Rightarrow \mathrm{y}\left({\mathrm{t}}_2+{\mathrm{t}}_1\right)-2{\mathrm{at}}_1{\mathrm{t}}_2-2{\mathrm{at}}_1^2=2\mathrm{x}-2{\mathrm{at}}_1^2\\ \Rightarrow 2\mathrm{x}-\left({\mathrm{t}}_2+{\mathrm{t}}_1\right)\mathrm{y}+2{\mathrm{at}}_1{\mathrm{t}}_2=0\end{array}$