Question

A variable plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ at a unit distance from origin cuts the coordinate axes at A, B and C. Centroid (x,y,z) satisfies the equation $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=K$ is

• A. 9
• B. 3
• C. $\frac{1}{9}$
• D. $\frac{1}{3}$

Answer 1

The correct option is A

9

Since $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ cuts the coordinate axes at $A(a,0,0),B(0,b,0),C(0,0,c).$
and its distance from origin = 1

$\frac{1}{\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}}=1or\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=1$
Where, P is centroid of triangle.
$\therefore P(x,y,z)=(\frac{a+0+0}{3},\frac{0+b+0}{3},\frac{0+0+c}{3})\Rightarrow x=\frac{a}{3},y=\frac{b}{3},z=\frac{c}{3}$
From Eqs. (i) and (ii) ,
$\frac{1}{9x^2}+\frac{1}{9y^2}+\frac{1}{9z^2}=1or\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=9=K\therefore K=9$