The points (-a, -b), (0, 0), (a, b) and ($a^2$, ab) are

Let $A(-a,-b),B(0,0),C(a,b)$ and $D(a^2,ab)$

Distance between two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ can be calculated using the formula $\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Distance between the points A$(-a,-b)$ and B $(0,0)=\sqrt{{\left(0+a\right)}^2+{\left(0+b\right)}^2}=\sqrt{a^2+b^2}$

Distance between the points B$(0,0)$ and C $(a,b)=\sqrt{{\left(a-0\right)}^2+{\left(b-0\right)}^2}=\sqrt{a^2+b^2}$

Distance between the points C$(a,b)$ and D $(a^2,ab)=\sqrt{{\left(a^2-a\right)}^2+{\left(ab-b\right)}^2}=(a-1)\sqrt{a^2+b^2}$

Distance between the points A$(-a,-b)$ and D $(a^2,ab)=\sqrt{{\left(a^2+a\right)}^2+{\left(ab+b\right)}^2}=(a+1)\sqrt{a^2+b^2}$

Now.

$AB+BC+CD=\sqrt{a^2+b^2}+\sqrt{a^2+b^2}+(a-1)\sqrt{a^2+b^2}=(a+1)\sqrt{a^2+b^2}=AD$

Hence, the points are collinear.