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

Question

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.

Try to find the slopes of the lines formed by joining each pair. — Kenny Lau, May 08 '16 at 06:32

score 3 · Answer 1 · answered May 08 '16 at 06:32

3

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?

answered May 08 '16 at 06:32

lab bhattacharjee

279,016

A quadratic can have atmost 2 roots. But, it has 3. Thank you very much – Swadhin May 08 '16 at 06:33
@Swadhin, Welcome! See https://www.quora.com/Can-a-quadratic-equation-have-more-than-2-roots – lab bhattacharjee May 08 '16 at 06:37

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

1 Answers1