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. $1/9$
• D. $1/3$

Answer 1

Answer: (A)

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 $\Delta ABC$ .
$\therefore$ $x=\frac{a}{3},y\frac{b}{3}andz=\frac{c}{3}$
$\Rightarrow$ $a=3x,b=3yc=3z$
On substituting the values of a, b and c in Eq. (i), we get
$\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=9$
$\therefore$ $k=9$