Show that the three distinct points $(p,p^2)$, $(q,q^2)$ and $(r,r^2)$ can never be collinear.

Question

Show that the three distinct points $(p,p^2)$, $(q,q^2)$ and $(r,r^2)$ can never be collinear.

I can think of the graph $y=x^2$ to solve the above problem graphically. However, I wanted to solve it mathematically, so I tried to find the area of the above 3 points and equate it to $0$. But, I couldn't do it. Please help.

Answer 1

Think of the equation of a straight line $$Ax+By+C=0$$

and if this passes through $(t,t^2),$

$$At+Bt^2+C=0\iff Bt^2+At+C=0$$ which is a Quadratic Equation in $t$ whose roots are $p,q,r$

How many roots a Quadratic Equation can have?