Question

A variable plane is at a constant distance p from the origin and meets the coordinate axes in points A, B and C respectively. Through these points, planes are drawn parallel to the coordinates planes. Find the locus of their point of intersection.

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

Answer 1

Answer: (D)

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

The equation of the plane is given by

$lx+my+nz=3p$

The intercepts on the 3 axes are $a=\frac{3p}{l},b=\frac{3p}{m},c=\frac{3p}{n}.$

The coordinates of the centroid are $x=\frac{a}{3}=\frac{p}{l};y=\frac{b}{3}=\frac{p}{m},z=\frac{c}{3}=\frac{p}{n}$

Now $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{l^2+m^2+n^2}{p^2}=\frac{1}{p^2}$.