Q25 of 47 Page 1

If a≠b≠0, prove that the points (a, a2), (b, b2) (0, 0) will not be collinear.

Let the points be A(a, a2), B(b, b2) and C(0, 0).

To prove: Points A, B, C are not collinear.


Three points are said to be collinear if the area of the triangle formed by the points is zero.


Area of a triangle when three points (x1, y1), (x2, y2), (x3, y3) are given:


Area of triangle = 1/2|[x1(y2 – y3) + x2(y3 - y1) + x3(y1 – y2)]|


∴ Area of ΔABC = 1/2 |[a(b2 - 0) + b(0 – a2) + 0(a2 – b2)]|


= 1/2|[ ab2 - ba2]|


= 1/2|[ab(b - a)]|


≠ 0 (b – a ≠ 0 ∵ a≠b≠0)


Therefore, points cannot be collinear.


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