Question

If a variable plane, at a distance of $3$ units from the origin, intersects the coordinate axes at $A, B$ and $C$, then the locus of the centroid of $\Delta $ABC is :

• A. $\frac{1}{x^{2}} + \frac{1}{y^{2} } + \frac{1}{z^{2}} = 1 $
• B. $\frac{1}{x^{2}} + \frac{1}{y^{2} } + \frac{1}{z^{2}} = 3 $
• C. $\frac{1}{x^{2}} + \frac{1}{y^{2} } + \frac{1}{z^{2}} = \frac{1}{9} $
• D. $\frac{1}{x^{2}} + \frac{1}{y^{2} } + \frac{1}{z^{2}} = 9 $

Answer 1

Answer: (A)

Let Centroid be $( h , k , l )$
$\therefore x -$ intp $=3 h \,\,\,\,\,Y$ -intp $=3 k ,\,\,\,\, 3$ -int $=3 l$
Equ. $\frac{x}{3 h}+\frac{y}{3 k}+\frac{z}{3 \ell}=1$
dist from $(0,0,0)$
$\left|\frac{-1}{\sqrt{\frac{1}{9 h^{2}}+\frac{1}{9 k^{2}}+\frac{1}{\left.9\right|^{2}}}}\right|=3$
$\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=1$