Question

A variable plane at a distance of $1$ unit from the origin cuts the coordinate axis at $A,B$ and $C$. If the centroid of triangle $ABC$ is $D(x,y,z)$ satisfy the relation $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=k$, then value of $k$ is

• A. $3$
• B. $1$
• C. $1/3$
• D. $9$

Answer 1

Answer: (D)

$9$

Let the plane be $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$

Distance of plane from origin is $\frac{1}{\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}}=1$

$⟹\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=1$

Centroid $D(x,y,z)=(\frac{a}{3},\frac{b}{3},\frac{c}{3})$

$a=3x,b=3y,c=3z$

$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=1$

$\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=9$