Question

If G(g), H(h) and (p) are centroid orthocentre and circumcentre of a triangle and xp + yh + zg = 0, then (x, y, z) is equal to ______.

Options

• 1, 1, –2

• 2, 1, –3

• 1, 3, –4

• 2, 3, –5

Answer 1

If G(g), H(h) and (p) are centroid orthocentre and circumcentre of a triangle and xp + yh + zg = 0, then (x, y, z) is equal to 2, 1, –3.

Explanation:

Since, the orthocentre, centroid and circumcentre of a triangle are collinear where as the centroid divides orthocentre and circumcentre in the ratio of 2 : 1.

By internal division formula, 

`(2p + 1h)/(2 + 1)` = g

`\implies` 2p + h – 3g = 0

As, x p + y h + z g = 0

Hence, x = 2, y = 1 and z = – 3