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 centroid 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 $\frac{16}{p^2}$

Let equation of plane is
lx+my+nz=p
or $\frac{x}{\left(\frac{p}{l}\right)}+\frac{y}{\left(\frac{p}{m}\right)}+\frac{z}{\left(\frac{p}{n}\right)}=1$
Coordinates of A, B, C are $(\frac{p}{l},0,0)(0,\frac{p}{m},0)$ and
$(0,0,\frac{p}{n})$ respectively.
$\therefore$ Centroid of OABC is $(\frac{p}{4l},\frac{p}{4m},\frac{p}{4n})$
$x_1=\frac{p}{4l},y_1=\frac{p}{4m},z_1=\frac{p}{4n}\because l^2+m^2+n^2=1\Rightarrow \frac{p^2}{16x_1^2}+\frac{p^2}{16y_1^2}+\frac{p^2}{16z_1^2}=1$
or $x_1^2{y}_1^2+y_1^2{z}_1^2+z_1^2{x}_1^2=16/p^2{x}_1^2{y}_1^2{z}_1^2$
$\therefore$ Locus is $x^2{y}^2+y^2{z}^2+z^2{x}^2=\frac{16}{p^2}{x}^2{y}^2{z}^2$
$\therefore k=\frac{16}{p^2}$