Question

If a unequal to b and b unequal to c then prove that the points (a,a2),(b,b2)(c,c2 can never be colinear)

Answer 1

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Answer 2

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 .