A plane meets the coordinate axes in points A, B, C and the centroid of the triangle ABC is . The equation of the plane is:

Question

A plane meets the coordinate axes in points A, B, C and the centroid of the triangle ABC is $(α, β, γ)$ . The equation of the plane is:

Smart Achievers · Accepted Answer

(1)

Ans. $\frac{x}{α} + \frac{y}{β} + \frac{z}{γ} = 3$

Explanation:

Let the equation of the required plane be

$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ ...(i)

It meets coordinate axes in points

$A (a, 0, 0), B (0, b, 0), C (0, 0, c)$

The centroid of $∆ A B C$ is $(\frac{a}{3}, \frac{b}{3}, \frac{c}{3})$

$\Rightarrow$ $\frac{a}{3} = α, \frac{b}{3} = β, \frac{c}{3} = γ$

$\Rightarrow$ $a = 3 α, b = 3 β, c = 3 γ$

Hence, the required plane is

$\frac{x}{3 α} + \frac{y}{3 β} + \frac{z}{3 γ} = 1$

i.e., $\frac{x}{α} + \frac{y}{β} + \frac{z}{γ} = 3$

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