Question

If the centroid of the triangle formed by the points (a, b), (1, a) and (b, 1) is at the origin, then `(a^3 + b^3 + 1)/(ab)` = ______.

Answer 1

If the centroid of the triangle formed by the points (a, b), (1, a) and (b, 1) is at the origin, then `(a^3 + b^3 + 1)/(ab)` = 3.

Explanation:

The centroid is `((a + 1 + b)/3, (b + a + 1)/3) = (0, 0)`, so a + b + 1 = 0.

Using the identity a³ + b³ + 1 – 3ab = (a + b + 1)(...), the left side equals 0 when a + b + 1 = 0, so a³ + b³ + 1 = 3ab. 

Therefore `(a^3 + b^3 + 1)/(ab) = 3`.