Question

If the centroid of the triangle formed by the points (a , b) , (b ,c) and (c , a) is at the origin , then $\left(\frac{a^3+b^3+c^3}{abc}\right)$ is

• A. 1
• B. 3
• C. 5
• D. -3

Answer 1

Answer: (B)

3

The centroid of the triangle is given by $(\frac{a+b+c}{3},\frac{a+b+c}{3})$

Given that the centroid of the triangle is origin.

$\Rightarrow (\frac{a+b+c}{3},\frac{a+b+c}{3})=(0,0)$

$\Rightarrow a+b+c=0$

Now consider the identity, $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$

Since $a+b+c=0$ we get

$a^3+b^3+c^3-3abc=0$

$\Rightarrow a^3+b^3+c^3=3abc$

$\Rightarrow \frac{a^3+b^3+c^3}{abc}=3$