Question

A variable plane which remains at a constant distance $3p$ from the origin, cuts the coordinates axes at $A,B$ and $C.$ Find the locus of the centroid of $\Delta$ $ABC.$

• A. $\frac{1}{x^2}$+$\frac{1}{y^2}$+$\frac{1}{z^2}=$ $\frac{1}{p^2}$
• B. $\frac{1}{x^4}$+$\frac{1}{y^4}$+$\frac{1}{z^4}=$ $\frac{1}{p^3}$
• C. $\frac{1}{x^4}$+$\frac{1}{y^4}$+$\frac{1}{z^4}=$ $\frac{1}{p^4}$
• D. None of these
  

Answer 1

Answer: (A)

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

Let the variable plane be $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$

$\therefore A=(a,0,0),B=(0,b,0),C=(0,0,c)$

Let $G(\alpha ,\beta ,\gamma )$ be the centroid of $△ABC$

$\therefore \alpha =\frac{a}{3},\beta =\frac{b}{3},\gamma =\frac{c}{3}$ .........$\left(1\right)$

Also given that, distance of plane from origin is $3p$

$\Rightarrow \frac{1}{\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}}=3p$

$\Rightarrow \frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{1}{9p^2}$

$\Rightarrow \frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}+\frac{1}{{\gamma }^2}=\frac{1}{p^2}$ using $\left(1\right)$

Hence, required locus of $G(\alpha ,\beta ,\gamma )$ is,

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