Question

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

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

Answer 1

Answer: (A)

$9$

Given plane cuts the coordinate axes at $A(a,0,0),B(0,b,0)$ and $C(0,0,c)$. It is at a unit distance from the origin.

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

$\Rightarrow \frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=1$ .........(i)

Since, $(x,y,z)$ is the centroid of $\Delta$ $ABC$.

$\therefore x=\frac{a}{3},y=\frac{b}{3}$ and $z=\frac{c}{3}$

$\Rightarrow a=3x,b=3y$ and $c=3z$

On substituting the values of a, b and c in Eq. (i), we get

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

$\Rightarrow k=9$