Question

If $a\ne b\ne c$ prove that $(a,a^2),(b,b^2)(0,0)$ will not be collinear.

Answer 1

three points be collinear when area of triangle formed by meeting of all three points = 0

The points A(a , a²) , B( b, b²) and C( c, c²) are given.
so, area of triangle ABC = [a( b² - c²) + b(c² - a²) + c(a² - b²)]
= [ab² - ac² + bc² - ba² + ca² - cb² ]
= [ ab(b - a) + bc(c - b) + ca(a - c)]

Here it is clear that area of triangle be zero when a = b = c . but a ≠ b ≠ c , so, area of triangle can't be zero. That's why all the given three points are never be collinear