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Question
If a≠b≠0, prove that the points (a, a2), (b, b2) (0, 0) will not be collinear.
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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 ∣∣`1/2`[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]∣∣ square units.
So,
Area of ∆ABC
`= |1/2|a(b^2 - 0) + b(0 - a^2) + 0(a^2 - b^2)||`
`=|1/2(ab^2 - a^2b)|`
`= 1/2|ab(b -a)|`
`!= 0 (∵ a!= b != 0)`
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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