Question : If the centroid of triangle formed by the points (a,b), (b,c) and (c,a) is at origin, then the value of a3+b3+c3 is
a) abc
b) 2abc
c) 3abc
d) none of these.
Doubt by Zoha
Solution :
If (x1, y1), (x2, y2) and (x3, y3) are the coordinates of the vertices of a triangle then the coordinates of the centroid are given by [(x1+x2+x3)/3 , (y1+y2+y3)/3]
Let
A (a,b)
B (b,c)
C (c,a)
Centroid G (0,0)
G(x,y) = [(x1+x2+x3)/3 , (y1+y2+y3)/3]
G(0,0) = [(a+b+c)/3 , (b+c+a)/3]
You can equate any x or y coordinate both sides. We are equating x-coordinate both sides.
0 = (a+b+c)/3
0×3 = a+b+c
0 = a+b+c
a+b+c = 0 — (1)
We know,
a3+b3+c3-3abc = (a+b+c)(a2+b2+c2-ab-bc-ca)
a3+b3+c3-3abc = (0)(a2+b2+c2-ab-bc-ca)
[Using equation (1)]
a3+b3+c3-3abc = 0
a3+b3+c3 = 3abc
Hence, c) would be the correct option.