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Question

If a ≠ b ≠ c, prove that (a, a2), (b, b2), (0, 0) will not be collinear. [CBSE 2017]

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Solution

Let A(a, a2), B(b, b2) and C(0, 0) be the coordinates of the given points.
We know that the area of triangle having vertices x1, y1, x2, y2 and x3, y3 is 12x1y2-y3+x2y3-y1+x3y1-y2 square units.
So,
Area of ∆ABC
=12ab2-0+b0-a2+0a2-b2=12ab2-a2b=12abb-a0 ab0
Since the area of the triangle formed by the points (a, a2), (b, b2) and (0, 0) is not zero, so the given points are not collinear.

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