Question

If a≠b≠c then prove that the points (a,a²),(b,b²) and (c,c²) can never be collinear

Answer 1

Method 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 = 1/2[a( b² - c²) + b(c² - a²) + c(a² - b²)]
= 1/2[ab² - ac² + bc² - ba² + ca² - cb² ]
= 1/2 [ 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.



method 2 :-
points A , B , C be collinear when ,
slope of AB = slope of BC = slope of CA

slope of AB = (b² - a²)/(b - a) = (b-a)(b + a)/(b-a)
= a + b
similarly,
slope of BC = (c² - b²)/(c - b) = b + c
slope of CA = c + a

here it is clear that ,
slope of AB ≠ slope of BC ≠ slope of CA
so, points A, B , C are never be collinear .