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If (a, b), (b, c) and (c, a) are the vertices of a triangle and the centroid of triangle is origin, then a + b + c = |
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Answer» Coordiante of the centroid of the triangle is:x=(x₁+x₂+x₃)/3 , y=(y₁+y₂+y₃)/3∴, According to the given question,(a+b+c)/3=0or, a+b+c=0 -----------------(1) Now, (a+b+c)³=a³+3a²b+3ab²+b³+3a²c+6abc+3b²c+3ac²+3bc²+c³or, (a+b+c)³=a³+b³+c³+3a²(b+c)+3b²(c+a)+3c²(a+b)+6abcor, a³+b³+c³=(a+b+c)³-3a²(-a)-3b²(-b)-3c²(-c)-6abc [Using (1)]or, a³+b³+c³=3a³+3b³+3c³-6abcor, a³+b³+c³-3a³-3b³-3c³=-6abcor, -2a³-2b³-2c³=-6abcor, a³+b³+c³=3abc Ans. |
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