The equation of the line passing through the points $(a,a^2),(b,b^2)$ is

Let $(x_1,y_1)=(a,a^2)$

Let $(x_2,y_2)=(b,b^2)$

Equation of two points is: $\frac{y-y_1}{y_1-y_2}=\frac{x-x_1}{x_1-x_2}$

$\frac{y-a^2}{a^2-b^2}=\frac{x-a}{a-b}$

$\frac{y-a^2}{(a+b)(a-b)}=\frac{x-a}{a-b}$

$\frac{y-a^2}{a+b}=x-a$

$y-a^2=(x-a)(a+b)$

$y-a^2=ax+bx-a^2-ab$

$y=ax+bx-ab$

$(a+b)x-y-ab=0$