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VITEEE 2016 Paper

Section: Mathematics

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Question : 107 of 125

Marks: +1, -0

A variable plane remains at constant distance p from the origin.If it meets coordinate axes at points A, B, C then the locus of the centroid of A ABC is

$x^{-2}+y^{-2}+z^{-2}$ = $9p^{-2}$
$x^{-3}+y^{-3}+z^{-3}$ = $9p^{-3}$
$x^{2}+y^{2}+z^{2}$ = $9p^{2}$
$x^{3}+y^{3}+z^{3}$ = $9p^{3}$

Solution:

Let A= (a, 0,0), B = (0, b, 0), C = (0,0, c), then equation of the plane is

\frac{x}{a}+\frac{y}{b}+\frac{z}{c}

= 1

Its distance from the origin,

\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}

=

\frac{1}{p^2}

... (i)

If (x, y, z) be centroid of δ ABC, then

x =

\frac{a}{3}

, y =

\frac{b}{3}

, z =

\frac{c}{3}

... (ii)

Eliminating a,b,c from (i) and (ii) required locus is

x^{-2}+y^{-2}+z^{-2}

=

9p^{-2}

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