Question

A variable plane which remains at a constant distance $p$ from the origin cuts the coordinate axes in $A,B,C$. The locus of the centorid of the tetrahedron $OABC$ is $y^2{z}^2+z^2{x}^2+x^2{y}^2=k{x}^2{y}^2{z}^2$, where $k$ is equal to

• A. $9p^2$
• B. $\frac{9}{p^2}$
• C. $\frac{7}{p^2}$
• D. $\frac{16}{p^2}$

Answer 1

The correct option is D $\frac{16}{p^2}$

A variable plane intersects the coordinate axes at $A(a,0,0),B(0,b,0),C(0,0,c)$.
Centroid of tetrahedron $OABC$ is $(x,y,z)\equiv (\frac{a}{4},\frac{b}{4},\frac{c}{4})$
Then equation of plane is $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
Perpendicular distance from origin to above plane,

$p=\left|\frac{-1}{\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}}\right|$

$\Rightarrow p^2=\frac{1}{\frac{1}{{16x}^2}+\frac{1}{16y^2}+\frac{1}{16z^2}}$

$\Rightarrow \frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{16}{p^2}$