If a≠b≠c,then prove that the points (a,a×a),(b,b×b), (c,c×c) can never be collinear.

If a≠b≠c,then prove that the points (a,a×a),(b,b×b), (c,c×c) ca

1.	If a≠b≠c,then prove that the points (a,a×a),(b,b×b), (c,c×c) can never be collinear.
Answer» Let A(a, a2), B(b, b2) and C(c, c2) be the given points.{tex}\\therefore{/tex}Area of\xa0{tex}\\Delta A B C{/tex}{tex}= \\frac { 1 } { 2 } \\left\\{ a \\left( b ^ { 2 } - c ^ { 2 } \\right) + b \\left( c ^ { 2 } - a ^ { 2 } \\right) + c \\left( a ^ { 2 } - b ^ { 2 } \\right) \\right\\}{/tex}{tex}= \\frac { 1 } { 2 } \\left\\{ a b ^ { 2 } - a c ^ { 2 } + b c ^ { 2 } - b a ^ { 2 } + c a ^ { 2 } - c b ^ { 2 } \\right\\}{/tex}{tex}= \\frac { 1 } { 2 } \\times 0{/tex} [if a = b = c]=0i.e., the points are collinear if a = b = cHence, the points can never be collinear if\xa0{tex}a \\neq b \\neq c.{/tex}