Plane ax + by + cz = 1 intersect axes in A,B,C respectively. If is a centroid of ΔABC then a+b+3c= ________

Engineering

Mathematics

Plane and Its Different Forms

Section Formulae and Centres of a Triangle

Distance from a Plane

Question

Plane ax + by + cz = 1 intersect axes in A,B,C respectively. If $G (\frac{1}{6}, - \frac{1}{3}, 1)$ is a centroid of ΔABC then a+b+3c= _________.

$\frac{4}{3}$

$\frac{5}{6}$

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let $A (\frac{1}{a}, 0, 0) B (0, \frac{1}{b}, 0) C (0, 0, \frac{1}{c})$
$centroid \Rightarrow (\frac{1}{3 a}, \frac{1}{3 b}, \frac{1}{3 c}) = (\frac{1}{6}, \frac{- 1}{3}, 1)$

On comparing we get,

3a = 6 ⇒ a = 23 b = −3

⇒ b = −1

$3 c = 1 \Rightarrow c = \frac{1}{3}$
$∴ a = 2, b = - 1, c = \frac{1}{3}$

∴ a + b + 3c = 2
Only (C) is correct.