If the centroid of the triangle formed by the point (a,b),(b,c) and (c,a) is at the origin then find the value of (a³+b³+c³).

• (A). abc
• (B). a+b+c
• (C). 3abc
• (D). 0

To find the value of \(a^3 + b^3 + c^3\) given that the centroid of the triangle formed by the points \((a,b)\), \((b,c)\), and \((c,a)\) is at the origin, we start by calculating the centroid.

Centroid Calculation

The formula for the centroid \(G\) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is:

G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)

Applying the Formula

For our points:

• \(x_1 = a\), \(y_1 = b\)
• \(x_2 = b\), \(y_2 = c\)
• \(x_3 = c\), \(y_3 = a\)

Substituting these into the centroid formula gives:

G = \left(\frac{a + b + c}{3}, \frac{b + c + a}{3}\right)

Setting the Centroid to the Origin

Since the centroid is at the origin, we have:

• \(\frac{a + b + c}{3} = 0\)
• \(\frac{b + c + a}{3} = 0\)

Both equations simplify to:

a + b + c = 0

Finding \(a^3 + b^3 + c^3\)

Using the identity for the sum of cubes, we know:

a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc)

Since \(a + b + c = 0\), the equation simplifies to:

a^3 + b^3 + c^3 = 3abc

Final Answer

Thus, the value of \(a^3 + b^3 + c^3\) is:

3abc

The correct option is (C) 3abc.