Triangles on a curve

Algebra Level 4

Consider the curve $y=$ $x^{3}$ Any three distinct points are chosen on this curve. The number of such triplets of points lying on this curve which form a triangle such that its centroid lies on the $y$ $axis$ is given by $P$ The value of P is

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Ankit Sultana
Mar 31, 2014

Consider points as $(a,a^{3})$ $(b,b^{3})$ and $(c,c^{3})$ . Now clearly $a+b+c=0$ for centroid to lie on y axis. Further for triangle to actually exist the determinant for area of triangle which will have the expansion $(a-b)(b-c)(c-a)(a+b+c)$ must be non zero. But clearly that can't be true for any real $a,b,c$ which satisfy $a+b+c=0$ . Thus no such triangle exists. Thus $\boxed{P=0}$

Triangles on a curve

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